\n Provides information of the type of a statement. This can be:
\n\n A mathematical problem (textbook/research level).\nIf this is a research problem then the user is also required to specify\nwhether the problem has already been solved.
\n\n An API statement
\n\n A \"test\" statement
\n\n The values of this attribute are
\n\n@[category textbook] : a textbook level math problem.
\n@[category research open] : an open reseach level math problem.
\n@[category research solved] : a solved reseach level math problem.\nThe criterion for being solved is that there exists an informal solution\nthat is widely accepted by experts in the area. In particular, this\ndoes
\n@[category test] : a statement that serves as a sanity check (e.g. for a new definition).
\n@[category API] : a statement that constructs basic theory around a new definition
\n Provides information about the existence of a formal proof for a statement.\nThis is independent of the category attribute and can be used with any category.
\n\n@[formal_proof using formal_conjectures at \"link\"] : formally proved in this repository.
\n@[formal_proof using lean4 at \"link\"] : formally proved in Lean 4 elsewhere.
\n@[formal_proof using other_system at \"link\"] : formally proved in another system\n(Roqc, Isabelle, Lean 3, HOL, etc.)
\n The tag should be used as follows:
\n@[category textbook]\ntheorem imo_2024_p6\n (IsAquaesulian : (ℚ → ℚ) → Prop)\n (IsAquaesulian_def : ∀ f, IsAquaesulian f ↔\n ∀ x y, f (x + f y) = f x + y ∨ f (f x + y) = x + f y) :\n IsLeast {(c : ℤ) | ∀ f, IsAquaesulian f → {(f r + f (-r)) | (r : ℚ)}.Finite ∧\n {(f r + f (-r)) | (r : ℚ)}.ncard ≤ c} 2 := by\n sorry\n\n@[category research open]\ntheorem an_open_problem : Transcendental ℝ (π + rexp 1) := by\n sorry\n\n@[category research solved, formal_proof using lean4 at \"https://example.com/proof\"]\ntheorem a_solved_problem_with_formal_proof : ... := by\n sorry\n\n@[category test]\ntheorem a_test_to_sanity_check_some_definition : ¬ FermatLastTheoremWith 1 := by\n sorry\n\n Provides information about the subject of a mathematical problem, via a\nnumeral corresponding to the AMS subject classification of the problem.\nThis can be used as follows:
\n@[AMS 11] -- 11 correponds to Number Theory in the AMS classification\ntheorem FLT : FermatLastTheorem := by\n sorry\n
\n The complete list of subjects can be found here:\nhttps://mathscinet.ams.org/mathscinet/msc/pdfs/classifications2020.pdf
\n\n In order to access the list from within a Lean file, use the #AMS command.
\n Note: the current implementation of the attribute includes all the main categories\nin the AMS classification for completeness. Some are not relevant to this repository.
","/FormalConjectures/Util/Attributes/AMS/":"\n This file defines some tools used by the ProblemSubject attribute in order classify\nproblems by their corresponding AMS Subject.
\n The AMSDescription has one term for each number n ∈ {1, ..., 96} that has a corresponding\nAMS subject, namely AMSDescription.«n». Note that not all values of n in this interval\nare assigned a subject.
\n To extract the value corresponding to n, one can use numToAMSDescriptions n. This is useful\nfor getting the doctring that corresponds to the subject n when parsing the attribute.
\n Finally, to access the list of subjects and their corresponding number when editing Lean files,\nwe implement a #AMS command that prints this list.
\n The AMSLinter is a linter to aid with formatting contributions to\nthe Formal Conjectures repository by ensuring that results in a file have\nthe appropriate subject tags.
\n This file contains test cases for the CategoryDocstringLinter, verifying that\nresearch-open, research-solved, and textbook declarations without docstrings are flagged,\nwhile declarations with docstrings or other categories are accepted.
\n This file implements a linter that checks that every file in the project\nhas the correct copyright header.
","/FormalConjectures/Util/Linters/ModuleDocstringLinter/":"\n This file implements a linter that enforces module docstring hygiene:
\n\nMissing module docstring: warns when a file has no /-! ... -/\nblock at all (detected on the first non-moduleDoc command).
\nDuplicate module docstrings: warns when a file has more than one\n/-! ... -/ block (subsequent blocks should be regular /- ... -/\ncomments).
\n The AnswerLinter is a linter to aid with using answer(sorry) correctly.
\n The CategoryDocstringLinter ensures that declarations tagged as\n@[category research open], @[category research solved], or @[category textbook] have a docstring.
\n The categoryLinter is a linter to aid with formatting contributions to\nthe Formal Conjectures repository by ensuring that results in a file have\nthe appropriate tags in order to distinguish between open/already solved\nproblems and background results/sanity checks.
\n The namespaceLinter is a linter to aid with formatting contributions to\nthe Formal Conjectures repository by ensuring that all declarations are\nplaced within a namespace.
\n Many misformalisations stem from using a pattern of the form ∃ x, P x → Q instead of\n∃ x, P x ∧ Q (e.g. when formalising something of the form \"there is positive x such that ...\").\nThis is almost always incorrect (and trivial to prove) since it then suffices to pick an x that\ndoes not satisfy P. This linter flags occurences of this patter to the user and proposes a\ncorrected syntax.
\n This file contains test cases for the AnswerLinter, verifying that it correctly flags\ntheorems with early arguments when answer(sorry) is the left-hand side of an iff,\nand does not flag theorems without answer(sorry) or without early arguments.
\n This file provides a standard set of imports used by problem files throughout the project.
","/FormalConjectures/Util/Answer/":"answer( ) elaborator\n This file provides syntax for marking up answers in a problem statement.
\n\n Note: certain problems also providing an answer, and can be formalised\nusing answer(sorry) as a placeholder. While providing a proof simply requires\nfinding any way to replace := sorry, providing an answer is not just finding\nany way to replace answer(sorry): it requires evaluation of mathematical meaning,\nwhich is a job for human mathematicians, not Lean alone.
\n Syntax definitions used in Google.Answer.
\n We separate them to allow handling the syntax without the full implementation.
","/FormalConjectures/Util/ForMathlib/":"\n This module is deprecated since 2026-01-08. Use FormalConjecturesForMathlib directly.
\n Provides the decl_name% term elaborator, which resolves an identifier to its fully qualified\nLean.Name at compile time. This ensures that references to declarations are checked by the\ncompiler and will cause build failures if the target declaration is renamed or removed.
\n The Hilbert–Smith conjecture states that a locally compact topological group acting\ncontinuously and faithfully on a connected finite-dimensional topological manifold must be a\nLie group. It remains open in general; Pardon proved it for 3-manifolds in 2013.\nAn equivalent formulation: no p-adic integer group ℤ_[p] can act faithfully on any\nconnected finite-dimensional topological manifold.
\n
\n Let $f(x_1, \\dots, x_n)$ be a multivariable polynomial with real coefficients that takes only\nnonnegative values for all real inputs.\nHilbert's 17th problem asks whether there exist rational functions $g_1, \\dots, g_m$ such that\n$f = g_1^2 + g_2^2 + \\cdots + g_m^2$. Resolved affirmatively by Artin in 1927.\n
\n Motzkin, \"The arithmetic-geometric inequality\". In Shisha, Oved (ed.). Inequalities. Academic Press. pp. 205–224.
\n\n
\narxiv/math.0110202\nA note on Banach--Mazur problem by
\nmathoverflow/41211\nEasy proof of the fact that isotropic spaces are Euclidean
\n\n
times p, times q conjectures\n
\n
\n
\n This file contains the formalisation of [GoLo21] up to and\nincluding Conjecture 1.8.
\n\n
\narxiv/1609.08688\nThe length of an $s$-increasing sequence of $r$-tuples by
\nGoLo21\nThe length of an $s$-increasing sequence of $r$-tuples\nby
\n
\n
\n
\n
\nSome Unconventional Problems in Number Theory\nby
\narxiv/2107.12475\nHardness of busy beaver value BB(15) by
\nHardness of Busy Beaver Value BB(15)\nby
\n
\n
\n
\n The Boxdot Conjecture was originally formulated by French and Humberstone and\nhas been studied in several works. In particular, see:
\n\n
\narxiv/1308.0994\nCluster Expansion and the Boxdot Conjecture by
\nThe Boxdot Conjecture and the Generalized McKinsey Axiom\nby
\n This file contains formal statements of some of the main open conjectures\nin complexity theory, including
\n\n the P vs NP problem
\n\n the NP vs coNP problem
\n\n
\n Arora, Sanjeev, and Boaz Barak. Computational complexity: a modern approach.\nCambridge University Press, 2009.
\n\n\n This file formalizes the Clay Mathematics Institute millennium problem concerning\nthe existence and smoothness of solutions to the Navier-Stokes equations in three\nspatial dimensions. While the definitions are generalized to arbitrary dimension n,\nthe millennium problem specifically concerns the case n = 3.
\n\n The Clay Millennium Problem asks for a proof of one of the following four statements:
\n\nnavier_stokes_existence_and_smoothness_R3: (A) Global existence on ℝ³
\nnavier_stokes_existence_and_smoothness_periodic: (B) Global existence on ℝ³/ℤ³
\nnavier_stokes_breakdown_R3: (C) Existence of breakdown scenario on ℝ³
\nnavier_stokes_breakdown_periodic: (D) Existence of breakdown scenario on ℝ³/ℤ³
\n Fefferman writes the velocity as $u(x,t)$, the initial velocity as $u^\\circ(x)$, the\npressure as $p(x,t)$, the force as $f(x,t)$, and the viscosity as $\\nu$. In Lean,\nu₀ : ℝ^n → ℝ^n denotes the initial velocity, while v : ℝ^n → ℝ → ℝ^n\ndenotes the solution velocity. The curried order v x t, p x t, and f x t\nkeeps the source convention that position comes before time.
\n Since the Clay statement gives equation (1) on the closed time half-line $t \\ge 0$,\nthe time derivative is encoded with derivWithin relative to Set.Ici 0. The Clay\nPDF also includes errata; in particular, we include spatial 1-periodicity of the\npressure in the periodic case. The sign correction to the weak-solution identity in\nthe errata is not represented here, since this file formalizes the four prize\nalternatives rather than the later weak-solution discussion.
\n The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function\n$\\zeta(s)$ have real part $\\frac{1}{2}$. The trivial zeros are the negative even integers\n$-2, -4, -6, \\ldots$. The hypothesis is one of the seven Millennium Prize Problems\nposed by the Clay Mathematics Institute.
\n\n The Generalized Riemann Hypothesis extends this to Dirichlet $L$-functions of primitive\nDirichlet characters.
\n\n Note: the Extended Riemann Hypothesis (ERH) for Dedekind zeta functions is intentionally\nnot stated here. Mathlib's NumberField.dedekindZeta is the naive Dirichlet series\n(LSeries), not a meromorphic continuation; outside the region of absolute convergence\ntsum returns junk 0, producing spurious zeros that make the naive foramlisation of the\nconjecture provably false. The ERH should be added once Mathlib provides a meromorphic\ncontinuation of the Dedekind zeta function.
\n
\nWikipedia: Riemann hypothesis
\n\nWikipedia: Generalized Riemann hypothesis
\n\nWikipedia: Dedekind zeta function
\n\n J. Neukirch,
\n D. A. Marcus,
\n References:
\n","/FormalConjectures/Mathoverflow/«75792»/":"\n Various questions about integer complexity, which is the minimum number of 1s needed to express a natural number using addition, multiplication, and parentheses.
\n Let ‖n‖ denote the integer complexity of n > 0.
\n It is known that ‖3^n‖ = 3n for n > 0.
\n Is it true that ‖2^n‖ = 2n for n > 0?
\n The corresponding conjecture for 5 is false, because\n5^6 = 15625 = 1 + 2^3 * 3^2 * (1 + 2^3 * 3^3)!
\n We have chosen to formalise this using an inductive type.
\n\n
\nmathoverflow/75792 by user Harry Altman
\n\n http://arxiv.org/abs/1203.6462 by Jānis Iraids, Kaspars Balodis, Juris Čerņenoks, Mārtiņš Opmanis, Rihards Opmanis, Kārlis Podnieks
\n\n http://arxiv.org/abs/1207.4841 by Harry Altman, Joshua Zelinsky
\n\n https://oeis.org/A5245 : Mahler-Popken complexity.
\n\n Why do polynomials with coefficients 0,1\nlike to have only factors with 0,1\ncoefficients?
\n\n
\n Does the 6-sphere $S^6$ admit the structure of a complex manifold?
\n\n
\n
\n Is there any polynomial $f(x, y) \\in \\mathbb{Q}[x, y]$ such that\n$f : \\mathbb{Q} \\times \\mathbb{Q} \\rightarrow \\mathbb{Q}$ is a bijection?
\n\n
\n
\n
\n
\n Can the unit square be covered by $1/k$-by-$1/(k+1)$ rectangles (across $1 \\le k$ natural)?
\n\n I am deliberately not requiring that the rotations can only be $0^\\circ, 90^\\circ, 180^\\circ, \\text{ or } 270^\\circ$.
\n\n Because of indexing, since n : ℕ starts at 0, we change the side lengths to $1 / (n + 1)$ and\n$1 / (n + 2)$, so that the first rectangle is $1/1$ by $1/2$, the second is $1/2$ by $1/3$, etc.
\n
\n Source:\nMathoverflow/31809
","/FormalConjectures/Mathoverflow/«10799»/":"\n
\nmathoverflow/10799\nasked by user
\nOptimal Monotone Families for the Discrete Isoperimetric Inequality\nby
\nAn Isoperimetric Inequality for the Hamming Cube and Integrality Gaps in Bounded-Degree\nGraphs by
\nA Proof of the Kahn–Kalai Conjecture by
\n Corollary 4.2 of Chapter 1 states that the sequence $(x^n), n = 1, 2, ... ,$ is equidistributed modulo 1 for\nalmost all x > 1. And a little bit further down:\n\"one does not know whether sequences such as $(e^n)$, $(π^n)$, or even $((\\frac 3 2)^n)$\"\nare equidistributed modulo 1 or not.
\n\n
\nUniform Distribution of Sequences\nby
\n
\n Borwein, J.; Bailey, D.; Girgensohn, R.
\n
\n [Bug12] Bugeaud, Yann. \"Distribution modulo one and Diophantine approximation.\"\nVol. 193. Cambridge University Press, 2012. Chapter 10.
\n\n [Men73] Mendès France, Michel. \"Les ensembles de Bésineau.\"\nSéminaire Delange-Pisot-Poitou 15.1 (1973): 1-6.
\n\n
\n [Bug12a] Bugeaud, Yann. \"Distribution modulo one and Diophantine approximation.\"\nVol. 193. Cambridge University Press, 2012. Chapter 10.
\n\n [Bug12b] Bugeaud, Yann, and Nikolay Moshchevitin. \"On fractional parts of powers\nof real numbers close to 1.\" Mathematische Zeitschrift 271.3 (2012): 627-637.
\n\n Chapter 10 of the book collects open questions. This file formalizes Problems 10.1,\n10.2, 10.3 and the unnumbered conjecture by Waldschmidt.
\n\n
\n [Bug12] Bugeaud, Yann. \"Distribution modulo one and Diophantine approximation.\"\nVol. 193. Cambridge University Press, 2012. Chapter 10.
\n\n [Har19] Hardy, Gr H. \"A problem of Diophantine approximation.\"\nJ. Indian Math. Soc 11 (1919): 162-166.
\n\n [Kok45] Koksma, J. F. \"Sur la théorie métrique des approximations diophantiques.\"\nIndag. Math 7 (1945): 54-70.
\n\n [Mah53] Mahler, Kurt. \"On the approximation of logarithms of algebraic numbers.\"\nPhilosophical Transactions of the Royal Society of London. Series A,\nMathematical and Physical Sciences 245.898 (1953): 371-398.
\n\nWal03\nWaldschmidt, Michel. \"Linear independence measures for logarithms of algebraic numbers.\"\nDiophantine Approximation: Lectures given at the CIME Summer School held in Cetraro, Italy,\nJune 28–July 6, 2000. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. 249-344.
\n\n The following problems were proposed and discussed by Dubickas as Conjecture 2 in [Dub09].
\n\n
\n [Bug12] Bugeaud, Yann. \"Distribution modulo one and Diophantine approximation.\"\nVol. 193. Cambridge University Press, 2012. Chapter 10.
\n\n [Dub09] Dubickas, Artūras. \"An approximation property of lacunary sequences.\"\nIsrael Journal of Mathematics 170.1 (2009): 95-111.
\n\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n [Ar25] Archivara Math Research Agent, An Additive Counterexample: Erdős Problem 897 (2025)
\n\n [ArWu25] Aristotle, operated mostly by L. Wu, Lean formalisation of Erdős problem 897 (2025)
\n\n [Wi70] E. Wirsing, A characterization of $\\log n$ as an additive arithmetic function.\nSymposia Math. (1970), 45-57.
\n\n [Wi81] E. Wirsing, Additive and completely additive functions with restricted growth.\nRecent progress in analytic number theory, Vol. 2 (Durham, 1979), 231--280 (1981).
\n\n
\n [Ad22] Adenwalla, S., Avoiding Monotone Arithmetic Progressions in Permutations of Integers.\narXiv:2211.04451 (2022).
\n\n [Ge19] Geneson, Jesse, Forbidden arithmetic progressions in permutations of subsets of the\nintegers. Discrete Math. (2019), 1489-1491.
\n\n
\n
\n Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords\nto $C_5$). Is it true that, if $H$ has $m$ edges and no isolated vertices, then\n$$ \\hat{r}(G,H) \\ll m? $$
\n\n In other words, is $G$ Ramsey size linear? A special case of Problem 566.
\n\n
\n [EFRS93] Erdős, Faudree, Rousseau and Schelp,
\n
\n
\n
\n
\n [Er56](Erdős, P., Problems and results in additive number theory.\nColloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 127-137.)
\n\n
\nMT25
\n\n
\n
\nChromatic cardinal: SimpleGraph.chromaticCardinal is the cardinal-valued chromatic number\ndefined in FormalConjecturesForMathlib. It extends the finite chromaticNumber (which takes\nvalues in ℕ∞) to a Cardinal, and is therefore able to distinguish between different infinite\nchromatic numbers.
\nTriangle-free subgraph: a subgraph H : G.Subgraph is triangle-free when H.coe.CliqueFree 3.\nThis is the standard Mathlib formulation: CliqueFree 3 means the graph has no K₃ as a clique.
\nSubgraph: we use G.Subgraph (a spanning subgraph record) rather than an induced subgraph\nsince the problem asks for any subgraph, not just induced ones.
\n
\n
\n
\n [Ta26] T. Tao, Erdős problem 358 (2026)
\n\n
\n
\n
\n [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.
\n\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\nTe19 G. Tenenbaum,\n
\n Is it true that there are no solutions to $n! = x^k \\pm y^k$ with $x,y,n \\in \\mathbb{N}$,\nwith $xy > 1$ and $k > 2$?
\n\n
\n [Br32] Breusch, Robert, Zur Verallgemeinerung des Bertrandschen Postulates, da\\ss zwischen $x$\nund 2 $x$ stets Primzahlen liegen. Math. Z. (1932), 505--526.
\n\n [ErOb37] Erdős, P. and Obláth, R., \"Über diophantische Gleichungen der Form $n!=x^p+y^p$ und\n$n!\\pmd m!=x^p$. Acta Litt. ac Sci. Reg. Univ. Hung. Fr.-Jos., Sect. Sci. Math. (1937), 241-255.
\n\n [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
\n\n [PoSh73] Pollack, Richard M. and Shapiro, Harold N., The next to last case of a factorial\ndiophantine equation. Comm. Pure Appl. Math. (1973), 313-325.
\n\n
\n
\n
\n
\n
\n
\n
\n
\n
\n [MoOs06] Moree, Pieter and Osburn, Robert, Two-dimensional lattices with few distances. Enseign. Math. (2) (2006), 361--380
\n\n [ErFi96] Erdős, Paul and Fishburn, Peter, Maximum planar sets that determine {$k$} distances. Discrete Math. (1996), 115--125.
\n\nGr26: Benjamin Grayzel, Solution to a Problem of Erdős Concerning Distances and Points
\n\n
\n [EES74] Ecklund, Jr., E. F. and Erd\\H{o}s, P. and Selfridge, J. L., A new function associated with\nthe prime factors of {$(\\sp{n}\\sb{k})$}. Math. Comp. (1974), 647--649.
\n\n [ELS93] Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor\nof a binomial coefficient. Math. Comp. (1993), 215--224.
\n\n [GrRa96] Granville, Andrew and Ramaré, Olivier, Explicit bounds on exponential sums and the\nscarcity of squarefree binomial coefficients. Mathematika (1996), 73--107.
\n\n [Ko99b] Konyagin, S. V., Estimates of the least prime factor of a binomial coefficient.\nMathematika (1999), 41--55.
\n\n [SSW20] Sorenson, Brianna and Sorenson, Jonathan and Webster, Jonathan, An algorithm and estimates\nfor the {E}rdős-{S}elfridge function. (2020), 371--385.
\n\n
\n
\n
\n
\n
\n [BrRo91] Brown, Tom C. and Rödl, Voijtech, Monochromatic solutions to equations with unit\nfractions. Bull. Austral. Math. Soc. (1991), 387-392.
\n\n
\n
\n [ErKo98] Erdős, P. and Komornik, V., Developments in non-integer bases.\nActa Math. Hungar. (1998), 57--83.
\n\n [Fe16] Feng, D.-J., On the topology of polynomials with bounded integer coefficients.\nJ. Eur. Math. Soc. (2016), 181--193.
\n\n
\n
\n
\n
\n [BaSc72] Barth, K. F. and Schneider, W. J., On a problem of Erd\\H{o}s concerning the zeros of the\nderivatives of an entire function. Proc. Amer. Math. Soc. (1972), 229--232.
\n\n [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.
\n\n
\n [Sc10] Schipperus, Rene, Countable partition ordinals. Ann. Pure Appl. Logic (2010), 1195-1215.
\n\n
\n Is the maximum size of a set $A \\subseteq {1, \\dots, N}$ such that $ab + 1$ is never\nsquarefree (for all $a, b \\in A$) achieved by taking those $n \\equiv 7 \\pmod{25}$?
\n\n
\n [Er92b] Erdős, P. \"Some of my favourite problems in number theory, combinatorics,\nand geometry.\" Resenhas do Instituto de Matemático e Estatística da Universidade\nde São Paulo 2.2 (1995): 165-186.
\n\n [Sa25] Sawhney, M. \"Problem 848.\" (2025)\nhttps://www.math.columbia.edu/~msawhney/Problem_848.pdf
\n\n Full formal proof of asymptotic result: https://github.com/The-Obstacle-Is-The-Way/erdos-banger
\n\n
\n
\n
\n [ESS94] Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number\nTheory (1994), 329-347.
\n\n
\n
\n
\n [ErSt75] Erdős, P. and Straus, E. G., Solution to Problem 387. Nieuw Arch. Wisk. (1975), 183.
\n\n [Sa75] Sattler, R., Solution to Problem 387. Nieuw Arch. Wisk. (1975), 184-189.
\n\n [Sa82b] Sattler, R., On Erdős property P₁ for the arithmetical sequence. Nederl. Akad. Wetensch.\nIndag. Math. (1982), 347--352.
\n\n [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\ntheory. Monographies de L'Enseignement Mathematique (1980).
\n\n [La26] D. Larsen, Erdős problem 318 (2026)
\n\n
\nA5244
\n\n [Ben Green's Open Problem 63](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 63)
\n\n
\n
\n
\n
\n
\n
\n
\n
\n
\n #TODO: Formalize the corresponding conjecture for infinite Sidon sets.
\n\n
\n [DM26a] DeepMind prover agent, formal proof of Erdős problem 152 (2026)
\n\n [DM26b] DeepMind prover agent, formal proof of the quadratic variant of Erdős problem 152 (2026)
\n\n [ESS94] Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number\nTheory (1994), 329-347.
\n\n
\n
\n [Ke60] Kesten, Harry, Uniform distribution {${\\rm mod},1$}. Ann. of Math. (2) (1960), 445--471.
\n\n
\n
\n
\n
\n [BoPi24] N. Bowler and M. Pitz, A note on uncountably chromatic graphs. arXiv:2402.05984 (2024).
\n\n [ErHa66] Erdős, P. and Hajnal, A., On chromatic number of graphs and set-systems. Acta Math. Acad.\nSci. Hungar. (1966), 61-99.
\n\n [Ko13] Komjáth, Péter, A note on chromatic number and connectivity of infinite graphs. Israel\nJ. Math. (2013), 499--506.
\n\n [So15] Soukup, Dániel T., Trees, ladders and graphs. J. Combin. Theory Ser. B (2015), 96--116.
\n\n [Th17] Thomassen, Carsten, Infinitely connected subgraphs in graphs of uncountable chromatic\nnumber. Combinatorica (2017), 785--793.
\n\n
\n [Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry.\nMath. Pannon. (1994), 261-269.
\n\n
\n [Gu04] Guy, Richard K.,
\n
\n
\n
\n [Ad25] S. Adenwalla, A Question of Erdős and Graham on Covering Systems. arXiv:2501.15170 (2025).
\n\n
\n [Ha68] Halász, G., Über die Mittelwerte multiplikativer zahlentheoretischer\nFunktionen. Acta Math. Acad. Sci. Hungar. (1968), 365-403.
\n\n [Wi67] Wirsing, E., Das asymptotische Verhalten von Summen über multiplikative Funktionen.\nActa Math. Acad. Sei. Hung. (1967), 411-467.
\n\n
\n
\n
\n
\n
\n
\n
\n
\n [Er65] Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.
\n\n [Er76e] Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282.
\n\nTang
\n\nTao
\n\n
\n [MRR19] J. Moreira, F.K. Richter, and D. Robertson, A proof of a sumset conjecture of Erdős,\nAnnals of Math. 189 (2019), 605-652.
\n\n
\n [BeFo99] Bezdek, Andr'{a}s and Fodor, Ferenc, Minimal diameter of certain sets in the plane. J. Combin. Theory Ser. A (1999), 105-111.
\n\n [Er94b] Erd\\H{o}s, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.
\n\n
\n
\n [Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6\n(1961), 221-254.
\n\n [ErHa71] Erdős, Paul and Hajnal, András, Unsolved problems in set theory. Axiomatic Set\nTheory, Proc. Sympos. Pure Math. XIII Part I (1971), 17-48.
\n\n [ErHa60] Erdős, Paul and Hajnal, András. On some combinatorial problems involving\ncomplete graphs. Acta Math. Acad. Sci. Hungar. (1960), 395-424.
\n\n [Gl62] Gladysz, S. Some topological properties of independent sets. Colloq. Math. (1962).
\n\n [He72] Hechler, S. H. A dozen small uncountable cardinals. TOPO 72, Lecture Notes\nin Math. (1972), 207-218.
\n\n [NPS87] Newelski, L., Pawlikowski, J., and Seredyński, F. Infinite independent sets in\nthe closed case. Acta Math. Acad. Sci. Hungar. (1987).
\n\n
\n [ErGy99] Erdős, Paul and Gyárfás, András, Split and balanced colorings of complete graphs.\nDiscrete Math. (1999), 79-86.
\n\n
\n
\n There are four possibilities for the density of $A+B$:
\n\n $A+B$ has zero upper and lower density (and hence also zero density).
\n\n $A+B$ has zero lower density, but positive upper density (and hence no density).
\n\n $A+B$ has positive upper and lower density that are equal (and hence positive density).
\n\n $A+B$ has positive upper and lower density that are unequal (and hence no density).
\n\n
\n
\n
\n
\n [BEGL96] Burr, S. A. and Erdős, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.
\n\n
\nSek59 Milan Sekanina, Замечания к фактoризации беcкoнечнoй цикличеcкoй группы, Czechoslovak Mathematical Journal, Vol. 9 (1959), No. 4, 485–495
\n\n
\n [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive\nintegers. Illinois J. Math. (1967), 428--430.
\n\n Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$\nvertices is the union of a graph with chromatic number $\\leq r$ and a graph with $\\leq f_r(m)$\nedges, then $G$ has chromatic number $\\leq r+1$.
\n\n Erdős asked whether:
\n\nf 2 n ≫ n
\n more generally, f r n ≫ r * n
\n This was disproved by Rödl, who constructed, for any $\\epsilon > 0$ and $k$, a graph with\nchromatic number $\\geq k$ such that every subgraph on $m$ vertices is bipartite after deleting at\nmost $\\epsilon m$ edges. This proves (in a strong sense) that $f_r(n) = o(n)$ for all fixed\n$r \\geq 2$.
\n\n
\n V. Rödl,
\n
\n [TaTe25] T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers. arXiv:2512.01739 (2025).
\n\n
\n
\n [Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138\n(2024).
\n\n [Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
\n\n [Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka\n(1992), 44-48.
\n\n
\n
\n
\n
\n [Ri76] Richter, Bernd, Über die Monotonie von Differenzenfolgen. Acta Arith. (1976), 225-227.
\n\n
\n
\n
\n
\n
\n
\n
\n [Da85] Danzer, L.,
\n
\n
\n [CoSc97] Corrales-Rodrigáñez, Capi and Schoof, René, The support problem and its\nelliptic analogue. J. Number Theory (1997) [Volume 64, Issue 2], 276--290.
\n\n
\n [MathOverflow] (https://mathoverflow.net/questions/138209/product-of-central-binomial-coefficients)
\n\n
\n
\n [APSSV26] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant,\nShort proofs in combinatorics and number theory.\narXiv:2603.29961 (2026).
\n\n [CLLW24] J. Champagne, T. Le, Y.-R. Liu, and T. D. Wooley, Well-distribution modulo one and the\nprimes. arXiv:2406.19491 (2024).
\n\n [Er64b] Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964),\n52-65.
\n\n [Er85e] Erdős, P., Some problems and results in number theory. Number theory and combinatorics.\nJapan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.
\n\n [Hl55] Hlawka, Edmund, Zur formalen {T}heorie der {G}leichverteilung in kompakten {G}ruppen. Rend.\nCirc. Mat. Palermo (2) (1955), 33--47.
\n\n [Mo26] P. Monticone, Lean formalisation of Erdős problem 997 (2026)
\n\n
\n
\n [BrLa99] Brown, Tom C. and Landman, Bruce M., Monochromatic arithmetic progressions with large\ndifferences. Bull. Austral. Math. Soc. (1999), 21--35.
\n\n
\n [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day\n(Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
\n\n [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math.\n(1980), 89-115.
\n\n
\n
\n
\n
\n
\n
\n [BKKKZ26] K. Barreto, J. Kang, S.-H. Kim, V. Kovač, and S. Zhang, Irrationality of rapidly\nconverging series: a problem of Erdős and Graham. arXiv:2601.21442 (2026).
\n\n [Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in\ntranscendence theory (Durham, 1986) (1988), 102-109.
\n\n [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\ntheory. Monographies de L'Enseignement Mathematique (1980).
\n\n [Fe26] T. Feng et al, Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős\nProblems. arXiv:2601.22401 (2026).
\n\n
\n
\n
\n
\n
\n
\n
\n [DeSz92] de Caen, D. and Székely, L. A., The maximum size of {$4$}- and {$6$}-cycle free bipartite\ngraphs on {$m,n$} vertices. (1992), 135--142.
\n\n [Er75] Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975),\n3-14.
\n\n [LUW94] Lazebnik, F. and Ustimenko, V. A. and Woldar, A. J., New constructions of bipartite graphs\non {$m,n$} vertices with many edges and without small cycles. J. Combin. Theory Ser. B (1994),\n111--117.
\n\n
\n
\n [GrYa68] Graver, Jack E. and Yackel, James, Some graph theoretic results associated with Ramsey's\ntheorem. J. Combinatorial Theory (1968), 125--175.
\n\n [MaVe23] Mattheus, S. and Verstraete, J., The asymptotics of $r(4,t)$. arXiv:2306.04007 (2023).
\n\n
\n [ErMi52] Erdős, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}. Proc. London Math. Soc. (3) (1952), 257--271.
\n\n
\n [Ch26] P. Chojecki and GPT-5.4 Pro, Erdős problem 258 (2026)
\n\n [St26] ster-oc, Lean formalisation of Erdős problem 258 (2026)
\n\n
\n [Po18] Pólya, Georg, Zur arithmetischen {U}ntersuchung der {P}olynome. Math. Z. (1918), 143--148.
\n\n [Wikipedia] https://en.wikipedia.org/wiki/Dickson%27s_conjecture
\n\n
\n
\n [ErSe83] Erdos, P. and Selfridge, J. L., Problem 6447. Amer. Math. Monthly (1983), 710.
\n\n [Gu04] Guy, Richard K.,
\n [Mo85] Monier (1985). No reference found.
\n\n
\n
\n Reviewed by @b-mehta on 2025-05-27
","/FormalConjectures/ErdosProblems/«283»/":"\n
\n [Gr63] Graham, R. L., A theorem on partitions. J. Austral. Math. Soc. (1963), 435-441.
\n\n
\n
\n
\n [Ch26] P. Chojecki, Bounded Representations by $x^2 + y^2 - z^2$ (2026)
\n\n [Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős\nand his mathematics\", Budapest, July 1999 (1999).
\n\n
\n
\n
\n
\nBorsuk's conjecture (1933): Is every bounded set of diameter 1 in $\\mathbb{R}^n$\nthe union of at most $n + 1$ sets of diameter strictly less than 1?
\n\n Erdős [Er44] suspected this is false for sufficiently large $n$. Confirmed\nby Kahn–Kalai [KK93], who disproved the conjecture for $n \\geq 2015$.\nThe current best is $n \\geq 64$ (Jenrich–Brouwer, 2014).
\n\n The conjecture is true for $n \\leq 3$ (Eggleston [Eg55] for $n = 3$).
\n\n [Bo33] Borsuk, K. (1933).
\n [Er44] Erdős, P. (1944). Remarks on a conjecture of Borsuk.
\n\n [Eg55] Eggleston, H. G. (1955).
\n [KK93] Kahn, J., Kalai, G. (1993).
\n Lean 4 code in this file was drafted with assistance from Claude (Anthropic).\nThe mathematical content and references are the author's own work.
","/FormalConjectures/ErdosProblems/«961»/":"\n
\n [Ju74] Jutila, Matti, On numbers with a large prime factor. {II}. J. Indian Math. Soc. (N.S.) (1974), 125--130.
\n\nRaSh73 Ramachandra, K. and Shorey, T. N., On gaps between numbers with a large prime factor. Acta Arith. (1973), 99--111.
\n\n
\n [Tao25] Tao, Terence. Sublevel Sets of Logarithmic Potentials. Terry Tao’s Blog, Dec. 2025\n(https://terrytao.wordpress.com/wp-content/uploads/2025/12/erdos-1038-1.pdf)
\n\n
\n
\n
\n
\n [Er48] Erdős, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.)\n(1948), 63-66.
\n\n
\n
\n
\n
\n
\n
\n
\n
\n
\n [ErMi52] Erdős, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}.\nProc. London Math. Soc. (3) (1952), 257--271.
\n\n [Sp81] Spiro, C. A., The frequency with which an integral-valued, prime-independent,\nmultiplicative or additive function of n divides a polynomial function of n.
\n\n [He84] Heath-Brown, D. R., The divisor function at consecutive integers.\nMathematika 31 (1984), no. 2, 141--149.
\n\n [Hi85] Hildebrand, A., The divisor function at consecutive integers. Pacific J. Math.\n(1987), 307--319
\n\n [EPS87] Erdős, P., Pomerance, C., and Sarkőzy, A., On locally repeated values of\narithmetic functions. III. Proc. Amer. Math. Soc. (1987), 1--7.
\n\n
\n [Pr26] D. Price and GPT-5.2 Pro, Erdős problem 851 (2026)
\n\n
\n [ClHa64] Clunie, J. and Hayman, W. K., The maximum term of a power series. J. Analyse Math.\n(1964), 143-186.
\n\n
\n
\n
\n
\n #TODO: Formalize the corresponding conjecture for infinite Sidon sets.
\n\n
\n [ESS94] Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number\nTheory (1994), 329-347.
\n\n
\n [Er75f]\nErdős, Paul, On some problems of elementary and combinatorial geometry.\nAnn. Mat. Pura Appl. (4) (1975), 99-108.
\n\n [Er83c]\nErdős, Paul, Combinatorial problems in geometry.\nMath. Chronicle (1983), 35-54.
\n\n [Er87b]\nErdős, P., Some combinatorial and metric problems in geometry.\nIntuitive geometry (Siófok, 1985) (1987), 167-177.
\n\n [Er90]\nErdős, Paul, Some of my favourite unsolved problems.\nA tribute to Paul Erdős (1990), 467-478.
\n\n [Er92b]\nErdős, Paul, Some of my favourite problems in various branches of combinatorics.\nMatematiche (Catania) (1992), 231-240.
\n\n [EFPR93]\nErdős, Paul and Füredi, Zoltán and Pach, János and Ruzsa, Imre Z.,\nThe grid revisited. Discrete Math. (1993), 189-196.
\n\n [Er94b]\nErdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry.\nMath. Pannon. (1994), 261-269.
\n\n [Er97e]\nErdős, Paul, Some of my favourite unsolved problems.\nMath. Japon. (1997), 527-537.
\n\n\n
\n [OD99] P. O'Donnell, High girth unit-distance graphs. PhD Dissertation, Rutgers University (1999).
\n\n
\n
\n For every 2-coloring of ℝ, is there an uncountable set $A ⊆ ℝ$ such that\nall sums $a + b$ for $a, b ∈ A, a ≠ b$ have the same colour?
\n\n
\n [Er75b] Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310.
\n\n [HLS17] Hindman, Neil and Leader, Imre and Strauss, Dona, Pairwise sums in colourings of the reals. Abh. Math. Semin. Univ. Hambg. (2017), 275--287.
\n\n [Ko16] Komjáth, Péter, A certain 2-coloring of the reals. Real Anal. Exchange (2016), 227--231.
\n\n [SWCol] Sokoup Dániel and Weiss, William, Sums and Anti-Ramsey Colourings of ℝ. https://danieltsoukup.github.io/academic/finset_colouring.pdf
\n\n
\n [Er49d] Erdös, P. \"On the strong law of large numbers.\" Transactions of the American Mathematical\nSociety 67.1 (1949): 51-56.
\n\n [Ma66] Matsuyama, Noboru. \"On the strong law of large numbers.\" Tohoku Mathematical Journal,\nSecond Series 18.3 (1966): 259-269.
\n\n
\n [Er80] Erdős, Paul,
\n
\n
\n
\n [ErHa68b] Erdős, P. and Hajnal, A., On chromatic number of infinite graphs. (1968), 83--98.
\n\n [Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.
\n\n
\n
\n
\n [Va39] van der Corput, J. G., Une in'egalit'e{} relative au nombre des diviseurs. Nederl. Akad. Wetensch., Proc. (1939), 547--553.
\n\n [Er52b] Erd\"os, P., On the sum {$\\sum^x_{k=1} d(f(k))$}. J. London Math. Soc. (1952), 7--15.
\n\n [Ho63] Hooley, Christopher, On the number of divisors of a quadratic polynomial. Acta Math. (1963), 97--114.
\n\n [Mc95] McKee, James, On the average number of divisors of quadratic polynomials. Math. Proc. Cambridge Philos. Soc. (1995), 389--392.
\n\n [Mc97] McKee, James, A note on the number of divisors of quadratic polynomials. (1997), 275--281.
\n\n [Mc99] McKee, James, The average number of divisors of an irreducible quadratic polynomial. Math. Proc. Cambridge Philos. Soc. (1999), 17--22.
\n\n [T] T. Tao, Erdos' divisor bound, https://terrytao.wordpress.com/2011/07/23/erdos-divisor-bound/
\n\n
\n
\n
\n [Fu63] Fuchs, W. H. J., Proof of a conjecture of G. Pólya concerning gap series. Illinois J.\nMath. (1963), 661--667.
\n\n [Ko65] Kövari, Thomas, A gap-theorem for entire functions of infinite order. Michigan Math. J.\n(1965), 133--140.
\n\n
\n [Bi28] Biernacki, Miécislas, Sur les équations algébriques contenant des paramétres arbitraires.\n(1928), 145.
\n\n
\nA54377 (Primary pseudoperfect numbers)
\n\n In this problem, a function $h : \\mathbb{N} \\to\\mathbb{N}$ is defined maximally by\nsome counting property.
\n\n The problem asks to estimate $h(n)$. This has been interpreted here as asking for $\\Theta(h(n))$.\nThe principal version includes answer(sorry) for an unknown function. On the other hand, the best\nknown upper bound is $\\sqrt{n}$ and the best known lower bound is $(n\\log(n))^{1/3}$ so we\nalso provide these candidates as variants. Moreover, it suffices to show $O(h(n))$ and\n$O((n\\log(n))^{1/3})$ respectively for each, so further variants are provided for those.
\n
\n [Str66] Straus, E. G.,
\n [Er62c] Erdős, Pál,
\n [Ch74b] Choi, S. L. G.,
\n
\n
\n [Po79] Pomerance, Carl, The prime number graph. Math. Comp. (1979), 399-408.
\n\n
\n This problem asks whether maximal Sidon sets can coexist with other Sidon sets that have\ndisjoint difference sets (apart from 0).
","/FormalConjectures/ErdosProblems/«442»/":"\n
\n
\n [BEFS89] Burr, S. A. and Erd\\H{o}s, P. and Faudree, R. J. and Schelp, R. H., On the difference\nbetween consecutive {R}amsey numbers. Utilitas Math. (1989), 115--118.
\n\n
\n
\n [Er34] Erdős, Paul, A Theorem of Sylvester and Schur. J. London Math. Soc. (1934), 282--288.
\n\n [Er55d] Erdős, P., On consecutive integers. Nieuw Arch. Wisk. (3) (1955), 124--128.
\n\n [Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.
\n\n
\n
\nA129515
\n\n
\n [BeEr74] Benkoski, S. J. and Erdős, P., On weird and pseudoperfect numbers. Math. Comp. (1974),\n617-623.
\n\n [HSS77] Hanson, F. and Steele, J. M. and Stenger, F., Distinct sums over subsets. Proc. Amer.\nMath. Soc. (1977), 179-180.
\n\n
\n
\nSt20 Steinhaus, Hugo, Sur les distances des points dans les ensembles de measure positive. Fund. Math. (1920), 93-104.
\n\n
\n
\n [erdosproblems.com/75] (https://www.erdosproblems.com/75)
\n\n
\n
\n
\n [ErRo97] Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.
\n\n
\n
\n
\n
\n [Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry.\nResenhas (1995), 165-186.
\n\n [Sch38] Schoenberg, I. J. \"On asymptotic distributions of arithmetical functions.\"\nTransactions of the American Mathematical Society 39.2 (1936): 315-330.
\n\n
\n [Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number\ntheory. Number theory (Eger, 1996) (1998), 169-180.
\n\n
\n [DoKo25] W. van Doorn and V. Kovač, Lacunary sequences whose reciprocal sums represent all\nrationals in an interval. arXiv:2509.24971 (2025).
\n\n
\n
\nA214583
\n\n [APSSV26b] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant,\nShort proofs in combinatorics, probability and number theory II.\narXiv:2604.06609 (2026).
\n\n [Or26] Y. Oriike, Lean formalisation of Erdős problem 1141 (2026)
\n\n [Po17] P. Pollack, Bounds for the first several prime character nonresidues. Proc. Amer. Math. Soc.\n(2017), 2815--2826.
\n\n [Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős\nand his mathematics\", Budapest, July 1999 (1999).
\n\n
\n [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\ntheory. Monographies de L'Enseignement Mathematique (1980).
\n\n [RST75] Ramachandra, K. and Shorey, T. N. and Tijdeman, R., On Grimm's problem relating to\nfactorisation of a block of consecutive integers. J. Reine Angew. Math. (1975), 109-124.
\n\n
\n
\n
\n
\n
\n
\n
\n
\n [CPZ23] Cohen, Alex, Cosmin Pohoata, and Dmitrii Zakharov. \"A new upper bound for the Heilbronn\ntriangle problem.\" arXiv preprint arXiv:2305.18253 (2023).
\n\n [CPZ24] Cohen, Alex, Cosmin Pohoata, and Dmitrii Zakharov. \"Lower bounds for incidences.\"\nInventiones mathematicae (2025): 1-74.
\n\n [KPS82] Komlós, János, János Pintz, and Endre Szemerédi. \"A lower bound for Heilbronn's problem.\"\nJournal of the London Mathematical Society 2.1 (1982): 13-24.
\n\n [KPS81] Komlós, János, János Pintz, and Endre Szemerédi. \"On Heilbronn's triangle problem.\"\nJournal of the London Mathematical Society 2.3 (1981): 385-396.
\n\n
\n
\n [ErGr80] Erdős, P. and Graham, R. L. (1980). Old and New Problems and Results in Combinatorial Number\nTheory. Monographies de L'Enseignement Mathématique, 28. Université de Genève. (See the\nsections on Egyptian fractions or practical numbers).
\n\n [Vo85] Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24.
\n\n
\n
\n [Er87b] Erdős, P., Some combinatorial and metric problems in geometry.\nIntuitive geometry (Siófok, 1985) (1987), 167-177.
\n\n [Ko24c] Z. Kovács, A note on Erdős's mysterious remark. arXiv:2412.05190 (2024).
\n\n\n
\n Reviewed by @b-mehta on 2025-05-27
","/FormalConjectures/ErdosProblems/«677»/":"\n
\n
\n [Er49d] Erdös, P. \"On the strong law of large numbers.\" Transactions of the American Mathematical\nSociety 67.1 (1949): 51-56.
\n\n [Ma66] Matsuyama, Noboru. \"On the strong law of large numbers.\" Tohoku Mathematical Journal,\nSecond Series 18.3 (1966): 259-269.
\n\n
\n
\n This is an \"eventually\" formulation of the Second Hardy–Littlewood conjecture.
","/FormalConjectures/ErdosProblems/«503»/":"\n
\n
\n [EFRS93] Erdős, Faudree, Rousseau and Schelp,
\n Let f_d(n) be minimal such that, in any set of n points in ℝ^d, there exist at most f_d(n) pairs\nof points which are distance 1 apart. Estimate f_d(n).
\n\n
\n
\n
\n
\n
\n
\n [Gu04] Guy, Richard K.,
\n
\nA039669
\n\n [Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős\nand his mathematics\", Budapest, July 1999 (1999).
\n\n [MiWe69] Mientka, W. E. and Weitzenkamp, R. C., On f-plentiful numbers, Journal of\nCombinatorial Theory, Volume 7, Issue 4, December 1969, pages 374-377.
\n\n
\n [ErRo97] Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith.\n(1997), 353--359.
\n\n
\n
\n
\n [Ha92] Hall, R. R., On some conjectures of Erdős in Astérisque. I. J. Number Theory (1992),\n313--319.
\n\n
\n [Ba75] Baumgartner, James E., Partitioning vector spaces. J. Combinatorial Theory Ser. A (1975),\n231-233.
\n\n
\n
\n
\n [ErSt71] Erdös, P., and E. G. Straus. \"Some number theoretic results.\" Pacific J. Math 36 (1971):\n635-646.
\n\n [ErSt74] Erdős, Paul, and Ernst Straus. \"On the irrationality of certain series.\" Pacific journal\nof mathematics 55.1 (1974): 85-92.
\n\n [ErKa54] P. Erdős, M. Kac, Amer. Math. Monthly 61 (1954), Problem 4518.
\n\n [ScPu06] Schlage-Puchta, J. C., The irrationality of a number theoretical series. Ramanujan J.\n(2006), 455-460.
\n\n [FLC07] Friedlander, J. B. and Luca, F. and Stoiciu, M., On the irrationality of a divisor\nfunction series. Integers (2007).
\n\n [Pr22] Pratt, K., The irrationality of a divisor function series of Erdős and Kac.\narXiv:2209.11124 (2022).
\n\n
\n
\n
\n
\n
\n
\n proven by considering the [Moser-Spindel graph]\nor the [Golomb graph]\n
\n
\n
\n [So09] Soifer, Alexander, How Does One Cut a Triangle? I
\n\n [So09c] Soifer, Alexander, Is there anything beyond the solution?
\n\n
\n [Bl21] Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726 (2021).
\n\n
\n
\n
\n
\n
\n
\n [Gr64d] Graham, R. L., A property of Fibonacci numbers. Fibonacci Quart. (1964), 1-10.
\n\n [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\ntheory. Monographies de L'Enseignement Mathematique (1980).
\n\n
\n For fixed n, k, let m(n, k) be minimal such that every family of subsets of [n]\nof size at least m(n, k) contains a k-sunflower.\nThe problem asks to estimate m(n, k), ideally asymptotically.
\n
\n
\n [Ru04] Ruzsa, Imre Z., Negative values of cosine sums. Acta Arith. (2004), 179-186.
\n\n [Be25c] B. Bedert, Polynomial bounds for the Chowla Cosine Problem. arXiv:2509.05260 (2025).
\n\n
\n
\n Let uₙ = pₙ / n, where pₙ is the nth prime. Does the set of n such that uₙ < uₙ₊₁\nhave positive density?
\n Erdős and Prachar also proved that ∑_{pₙ < x} |uₙ₊₁ - uₙ| ≍ (log x)^2, and that the set of n\nsuch that uₙ > uₙ₊₁ has positive density. Erdős also asked whether there are infinitely many\nincreasing triples uₙ < uₙ₊₁ < uₙ₊₂ or decreasing triples uₙ > uₙ₊₁ > uₙ₊₂.
\n
\n [ErPr61] Erdős, P. and Prachar, K.,
\n
\n [He88] Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. (1988)
\n\n
\n
\n
\n [Er87] Erdős,
\n [NeRo75] Nešetřil and Rödl,
\n
\n
\n [Si56] Sierpiński, W., Sur les décompositions de nombres rationnels en fractions primaires.\nMathesis (1956), 16--32.
\n\n
\n
\n
\n [ErGr80] Erdős, P. and Graham, R. L., Old and New Problems and Results in Combinatorial\nNumber Theory. Monographie de l'Enseignement Mathématique, No. 28 (1980).
\n\n [Si60] Sierpiński, W., Elementary Theory of Numbers. Państwowe Wydawnictwo Naukowe,\nWarsaw (1960).
\n\n [FFK08] Filaseta, M., Finch, C., and Kozek, M., On powers associated with Sierpiński numbers,\nRiesel numbers and Polignac's conjecture. Journal of Number Theory 128 (2008), 1916–1940.
\n\n A positive odd integer $k$ is a
\n Sierpiński (1960) proved that infinitely many Sierpiński numbers exist using covering systems.\nThe smallest known Sierpiński number is 78557 (Selfridge). Erdős and Graham conjectured that\nthere exist Sierpiński numbers with no finite covering set. A negative answer would imply\ninfinitely many Fermat primes.
\n\n Note: The notion of a covering set for a Sierpiński number is closely related to a\nCoveringSystem of $\\mathbb{Z}$ (see\nFormalConjecturesForMathlib.NumberTheory.CoveringSystem): a finite covering set of primes\nfor $k$ works because the exponents $n$ for which each prime divides $k \\cdot 2^n + 1$ form\nresidue classes whose union covers all of $\\mathbb{Z}$, i.e. a covering system.
\n See also Erdős Problems 203 and\n276.
","/FormalConjectures/ErdosProblems/«930»/":"\n
\n
\n
\n
\n [LaLa26] Larsen and Larsen, Erdős problem 868 (2026)
\n\n
\n [Er82c] Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.
\n\n
\n [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\ntheory. Monographies de L'Enseignement Mathematique (1980).
\n\n [ErHa78] Erdős, P. and Hall, R. R., On some unconventional problems on the divisors of integers.\nJ. Austral. Math. Soc. Ser. A (1978), 479--485.
\n\n The Collatz conjecture states that for any positive integer $n$, there exists a natural\nnumber $m$ such that the $m$-th term of the sequence is 1.
\n\n
\n [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
\n\n [La10] Lagarias, Jeffrey C., The {$3x+1$} problem: an overview. (2010), 3--29.
\n\n [La16] Lagarias, Jeffrey C., Erdős, Klarner, and the {$3x+1$} problem. Amer. Math. Monthly\n(2016), 753--776.
\n\n [La85] Lagarias, Jeffrey C., The {$3x+1$} problem and its generalizations. Amer. Math. Monthly\n(1985), 3--23.
\n\n This file points to the canonical formalization in\nFormalConjectures.Wikipedia.CollatzConjecture.
\n
\n
\n
\n [He00] Heath-Brown, D. R., Arithmetic applications of {K}loosterman sums. Nieuw Arch. Wiskd. (5)\n(2000), 380--384.
\n\n\n
\n
\n
\n
\n
\n [Ch72] Chang, C. C., A partition theorem for the complete graph on {$\\omega\\sp{\\omega }$}. J. Combinatorial Theory Ser. A (1972), 396-452.
\n\n [Sp57] Specker, Ernst, Teilmengen von Mengen mit Relationen. Comment. Math. Helv. (1957), 302-314.
\n\n [La73] Larson, Jean A., A short proof of a partition theorem for the ordinal {$\\omega \\sp{\\omega }$}. Ann. Math. Logic (1973/74), 129-145.
\n\n
\n
\nA5236
\n\n Erdős called a natural number n a ω, the number of distinct prime divisors,\nif m + ω(m) ≤ n for all m < n. He believed there should be infinitely many such barriers, and\neven posed a relaxed variant asking whether there is some ε > 0 for which infinitely many n\nsatisfy m + ε · ω(m) ≤ n for every m < n.
\n
\n [Be68] Berlekamp, E. R., A construction for partitions which avoid long arithmetic progressions. Canad. Math. Bull. (1968), 409-414.
\n\n [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
\n\n [Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
\n\n [Go01] Gowers, W. T., A new proof of Szemerédi's theorem. Geom. Funct. Anal. (2001), 465-588.
\n\n
\n [EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J.\nAnalyse Math. (1958), 125-148.
\n\n [Po59] Pommerenke, Ch., On some problems by Erdős, Herzog and Piranian. Michigan Math. J.\n(1959), 221-225.
\n\n [Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961),\n97-115.
\n\n
\n
\n [Kanold](No references found)
\n\n [GuKa15](Guth, Larry and Katz, Nets Hawk, On the Erd\\H{o}s distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.)
\n\n [Piepmeyer](No references found)
\n\n
\n
\n
\n
\n [GIL24] Gabdullin, Mikhail R. and Iudelevich, Vitalii V. and Luca,\nFlorian, Numbers of the form {$k+f(k)$}. J. Number Theory (2024), 58--85.
\n\n
\n [CrVE70] R.B. Crittenden and C.L. Vanden Eynden,
\n
\n
\n
\n
\n
\n [Sa97] Sándor, Csaba, On a problem of Erdős. J. Number Theory (1997), 203-210.
\n\n
\n
\n [Hi79] Hindman, Neil, Partitions and sums of integers with repetition.\nJ. Combin. Theory Ser. A (1979), 19--32.
\n\n [Ow74] J. Owings, E2494. Amer. Math. Monthly (1974), 902.
\n\n
\n [ABJ11] Ardal, H. and Brown, T. and Jungić, V., Chaotic orderings of the rationals and reals. Amer. Math. Monthly (2011), 921-925.
\n\n
\n [KoLu25] V. Kovač and F. Luca, On the number of divisors of Mersenne numbers. arXiv:2506.04883 (2025).
\n\n
\n [EGMRSS75] Erdős, P. and Graham, R. L. and Montgomery, P. and Rothschild, B. L. and Spencer, J.\nand Straus, E. G., Euclidean {R}amsey theorems. {II}. (1975), 529--557.
\n\n [Ts17] Tsaturian, Sergei, A {E}uclidean {R}amsey result in the plane. Electron. J. Combin. (2017),\nPaper No. 4.35, 9.
\n\n References:
\n\n [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\ntheory. Monographies de L'Enseignement Mathematique (1980).
\n\n [GeRa79] Gerver, Joseph L. and Ramsey, L. Thomas, \"On certain sequences of lattice points.\"\nPacific J. Math. (1979), 357-363.
\n\n
\n
\n [CES75] Choi, S. L. G. and Erdős, P. and Szemerédi, E., Some additive and multiplicative problems\nin number theory. Acta Arith. (1975), 37--50.
\n\n
\n
\n
\n [SeSt58] Selfridge, J. L. and Straus, E., On the determination of numbers by their sums\nof a fixed order. Pacific Journal of Math. (1958), 847-856.
\n\n [Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n221-254.
\n\n [GFS62] Gordon, B. and Fraenkel, A. S. and Straus, E. G., On the determination of sets\nby the sets of sums of a certain order. Pacific J. Math. (1962), 187--196.
\n\n
\n
\n
\n [Er94b] Erdős, Paul,
\n
\n
\n
\n
\n
\n [EGH75] Erdős, Paul and Galvin, Fred and Hajnal, András, On set-systems having large\nchromatic number and not containing prescribed subsystems.\nInfinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th\nbirthday), Vol. I. Colloq. Math. Soc. János Bolyai 10, North-Holland (1975), 425–513.
\n\n [Er95d] Erdős, Paul, Some of my favourite problems in various branches of combinatorics.\nMatematiche (Catania) 47 (1992), no. 2, 231–240 (1995).
\n\n
\n
\n
\n [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
\n\n [Fa66] Faulkner, M. \"On a theorem of Sylvester and Schur.\" Journal of the London Mathematical\nSociety 1.1 (1966): 107-110.
\n\n
\n [Er87] Erdős, Paul, Problems and results on set systems and hypergraphs. Extremal problems\nfor finite sets (Visegrád, 1991), Bolyai Soc. Math. Stud. (1994), 217-227.
\n\n [Fo70] Folkman, Jon, Graphs with monochromatic complete subgraphs in every edge coloring.\nSIAM J. Appl. Math. (1970), 19:340-345.
\n\n [NeRo75] Nešetřil, Jaroslav and Rödl, Vojtěch, Type theory of partition problems of graphs.\nRecent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974),\nAcademia, Prague (1975), 405-412.
\n\n
\n In this problem, a function $h : \\mathbb{N} \\to\\mathbb{N}$ is defined maximally by a specified\ncounting property.
\n\n The problem asks to estimate $h(n)$. This has been interpreted here as asking for $\\Theta(h(n))$.\nThe principal version includes answer(sorry) for an unknown function. On the other hand, the best\nknown upper bound is $n^{2/3}$ and the best known lower bound is $\\sqrt{n}$ so we\nalso provide these candidates as variants. Moreover, it suffices to show $O(h(n))$ and\n$O(\\sqrt{n})$ respectively for each, so further variants are provided for those.
\n In the source paper [Er73], Erdős also remarks that it should not be too difficult\nto determine $\\lim_{n\\to\\infty}\\log(h(n))/\\log(n)$. This does not appear on the website, and\nit is not clear whether this remains open, but we include it here either way.
\n\n
\n [GR99] Granville, A., & Roesler, F. (1999).
\n [Er73] Erdős, P.,
\n
\n
\n [EGS82] Erdős, P., R. K. Guy, and J. L. Selfridge. \"Another Property of 239 and some related\nquestions.\" Congr. Numer. 34 (1982): 243-257.
\n\n
\n
\n
\n
\n [BFT15] Banks, William D. and Freiberg, Tristan and Turnage-Butterbaugh, Caroline L., Consecutive primes in tuples. Acta Arith. (2015), 261-266.
\n\n [Ma15] Maynard, James, Small gaps between primes. Ann. of Math. (2) (2015), 383-413.
\n\n
\n
\narxiv/2510.19804 Boris Alexeev and Dustin G. Mixon, Forbidden\nSidon subsets of perfect difference sets, featuring a human-assisted proof (2025)
\n\n [Ha47] Marshall Hall, Jr., Cyclic projective planes, Duke Math. J. 14 (1947), 1079–1090.
\n\n Let A ⊆ ℕ be a finite Sidon set. Is there some set B with A ⊆ B which is a perfect\ndifference set modulo p^2 + p + 1 for some prime power p?
\n This problem is related to Erdős Problem 329 about the maximum density of Sidon sets.\nIf this conjecture is true, it would imply that the maximum density of Sidon sets is 1.
","/FormalConjectures/ErdosProblems/«359»/":"\n
\n
\n
\n
\n [Er64] P. Erdős,
\n [AbHa70] H. L. Abbott and D. Hanson,
\n [Er73] P. Erdős,
\n
\n [Bl21] Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726 (2021).
\n\n
\n
\n [RRS24] Reiher, Christian and R\"odl, Vojt\\v ech and Sales, Marcelo, Colouring versus density in integers and {H}ales-{J}ewett cubes. J. Lond. Math. Soc. (2) (2024)\narXiv:2311.08556
\n\n
\n
\n
\n [BoCh62] Bose, R. C. and Chowla, S., Theorems in the additive theory of numbers. Comment. Math.\nHelv. (1962/63), 141-147.
\n\n [Gr01] Green, Ben, The number of squares and {$B_h[g]$} sets. Acta Arith. (2001), 365-390.
\n\n [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
\n\n
\n
\n
\n [KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series.
\n\n
\n
\n
\n
\n [Ch23] Chen, Yong-Gao, A conjecture of Erdős on $p+2^k$. arXiv:2312.04120 (2023).
\n\n [Er50] Erdős, P., On integers of the form $2^k+p$ and some related problems. Summa Brasil. Math.\n(1950), 113-123.
\n\n [Ro34] Romanoff, N. P., Über einige Sätze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.
\n\n
\n
\n
\n
\n Let f_2(n) be the maximum number of pairs of points at distance exactly 1\namong any set of n points in ℝ², under the condition that all pairwise\ndistances are at least 1.
\n Estimate the growth of f_2(n).
\n Status: open.
","/FormalConjectures/ErdosProblems/«13»/":"\n
\n
\n
\n [ESS75] Erdős, P. and Simonovits, M. and Sós, V. T., Anti-{R}amsey theorems. (1975), 633--643.
\n\n [MoNe05] Montellano-Ballesteros, J. J. and Neumann-Lara, V., An anti-{R}amsey theorem on cycles.\nGraphs Combin. (2005), 343--354.
\n\n [SiSo84] Simonovits, Miklós and Sós, Vera T., On restricted colourings of {$K_n$}. Combinatorica\n(1984), 101--110.
\n\n [Yu21] L.-T. Yuan, The anti-Ramsey number for paths. arXiv:2102.00807 (2021).
\n\n
\n [ErSz76] Erdős, E. and Szemerédi, E., On a problem of Graham. Publ. Math. Debrecen (1976),\n123--127.
\n\n [GHW10] Gao, Weidong and Hamidoune, Yahya Ould and Wang, Guoqing, Distinct length modular zero-sum\nsubsequences: a proof of Graham's conjecture. J. Number Theory (2010), 1425--1431.
\n\n
\n [Bo99] Bourgain, J., On the dimension of {K}akeya sets and related maximal\ninequalities. Geom. Funct. Anal. (1999), 256--282
\n\n [KaTa99] Katz, Nets Hawk and Tao, Terence, Bounds on arithmetic projections, and applications to the\n{K}akeya conjecture. Math. Res. Lett. (1999), 625--630.
\n\n [Le15] Lemm, Marius, New counterexamples for sums-differences. Proc. Amer. Math. Soc. (2015), 3863--3868.
\n\n [GGTW25] B. Georgiev, J. Gómez-Serrano, T. Tao, and A. Wagner, Mathematical exploration and discovery at scale. arXiv:2511.02864 (2025).
\n\n
\n [ErRo97] Erdős, P. and Rosenfeld, M., The factor-difference set of integers. (1997)
\n\n [Ji99] Jiménez-Urroz, J., A note on a conjecture of Erdős and {R}osenfeld. (1999)
\n\n [Br19] Bremner, A., On a problem of Erdős related to common factor differences. (2019)
\n\n
\n [Erd54] Erdős, Paul, Some results on additive number theory. Proc. Amer. Math. Soc. (1954),\n847-853.
\n\n [Guy04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437
\n\n [Ru98c] Ruzsa, Imre Z., On the additive completion of primes. Acta Arith. (1998), 269-275.
\n\n
\n
\n
\n [GyLe95] Gyárfás, András and Lehel, Jenő, Linear sets with five distinct differences among any\nfour elements. J. Combin. Theory Ser. B (1995), 108-118.
\n\n
\n
\n
\n
\n
\n
\n
\n [Ta16] Tao, Terence, The Erdős discrepancy problem. Discrete Anal. (2016), Paper No. 1, 29.
\n\n
\n
\n [Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken\nfrom a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.
\n\n
\n [BM22] Bucić, M. and Montgomery, R., Towards the Erdős-Gallai Cycle Decomposition Conjecture.\narXiv:2211.07689 (2022).
\n\n [CFS14] Conlon, David and Fox, Jacob and Sudakov, Benny, Cycle packing. Random Structures\nAlgorithms (2014), 608-626.
\n\n [EGP66] Erdős, Paul and Goodman, A. W. and Pósa, Lajos, The representation of a graph by set\nintersections. Canadian J. Math. (1966), 106-112.
\n\n [Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial\nMathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.
\n\n
\n
\nA74741
\n\n\n
\n
\n [ErLe96] Erdős, P. and Lewin, Mordechai,
\n [Er92b] Erdős, Paul,
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n [BaLu05] Banks, William D. and Luca, Florian, Nonaliquots and {R}obbins numbers. Colloq. Math.\n(2005), 27--32.
\n\n [BrSc95] Browkin, J. and Schinzel, A., On integers not of the form {$n-\\phi(n)$}. Colloq. Math.\n(1995), 55-58.
\n\n [ChZh11] Chen, Yong-Gao and Zhao, Qing-Qing, Nonaliquot numbers. Publ. Math. Debrecen (2011),\n439--442.
\n\n [Er73b] Erdős, P., \"Über die Zahlen der Form $\\sigma (n)-n$ und $n-\\phi(n)$. Elem. Math.\n(1973), 83-86.
\n\n [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
\n\n [PoPo16] Pollack, Paul and Pomerance, Carl, Some problems of Erdős on the sum-of-divisors\nfunction. Trans. Amer. Math. Soc. Ser. B (2016), 1-26.
\n\n
\n
\n
\n
\n
\n
\n
\n
\n [Ho67] Hooley, C., On the power free values of polynomials. Mathematika (1967), 21--26.
\n\n [Br11] Browning, T. D., Power-free values of polynomials. Arch. Math. (Basel) (2011), 139--150.
\n\n [Er53] Erdős, P., Arithmetical properties of polynomials. J. London Math. Soc. (1953), 416--425.
\n\n
\n [Ha79] Haight, J. A., Covering systems of congruences, a negative result. Mathematika (1979),\n53--61.
\n\n A Steiner system $S(t, k, n)$ is a collection of $k$-element subsets (called blocks) of\nan $n$-element set such that every $t$-element subset is contained in exactly one block.
\n\n
\nLarge Steiner Systems\nby Kunal Marwaha
\n\n A synchronizing word (also called a reset word) for a deterministic finite automaton (DFA)\n$M = (Q, \\Sigma, \\delta)$ is a word $w \\in \\Sigma^
\n A DFA is called synchronizing if it admits at least one synchronizing word.
\n\n The Černý conjecture asserts that every synchronizing DFA with $n$ states has a\nsynchronizing word of length at most $(n - 1)^2$. This bound is sharp: the family of Černý\nautomata $C_n$ witnesses it, requiring exactly $(n - 1)^2$ steps.
\n\nStatus: Open. The best known upper bound is\n$\\left(\\frac{7}{48} + \\frac{2 \\cdot 15625}{1597536}\\right) n^3 + o(n^3) \\approx 0.1654,n^3$\n(Shitov, 2019). The bound $(n - 1)^2$ has been verified for small $n$ and for special classes of\nautomata (e.g., Eulerian, aperiodic, cyclic automata).
\n\n We use Mathlib's DFA α σ (from Mathlib.Computability.DFA), together with the auxiliary\nDFA.IsSynchronizingWord and DFA.IsSynchronizing predicates defined in\nFormalConjecturesForMathlib.Computability.DFA.
\n
\nWikipedia: Synchronizing word
\n\n J. Černý,
\n Y. Shitov,
\n
\n
\n The Pierce-Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed\nas a maximum of finite minima of finite collections of polynomials. It was first stated in 1956\nby Garrett Birkhoff and Richard S. Pierce, though the modern rigorous formulation is due to\nMelvin Henriksen and John R. Isbell.
\n\n The conjecture has been proved for n = 1 and n = 2 by Louis Mahé.
\n
\n [Wikipedia] (https://en.wikipedia.org/wiki/Weird_number)
\n\n [A006037] (https://oeis.org/A006037)
\n\n
\n
\n
\n
\n Every positive integer is the sum of at most 5 tetrahedral numbers.
\n\n
\nA797
\n\n L. E. Dickson,
\n Frederick Pollock,
\n H. E. Salzer and N. Levine,
\n
\n
\n
\nWikipedia,
\nH. A. Helfgott and Á. Seress,
\n This file contains two conjectures from the Babai–Seress paper:
\n\nConjecture 1.5: $\\operatorname{diam}(A_n) < n^C$ for some absolute constant $C$,\nwhere $A_n$ is the alternating group on $n$ elements.
\n\nConjecture 1.7: $\\operatorname{diam}(G) < (\\log |G|)^C$ for some absolute constant $C$,\nwhere $G$ ranges over all non-abelian finite simple groups.
\n\n Conjecture 1.7 generalises Conjecture 1.5, since for $G = A_n$ we have\n$\\log |A_n| \\approx n \\log n$, so a polylogarithmic bound in $|G|$ implies a polynomial\nbound in $n$.
","/FormalConjectures/Wikipedia/RamseyNumbers/":"\n The (graph) Ramsey number $R(k,\\ell)$ is the least natural number $n$ such that every simple graph\non $n$ vertices contains either a clique of size $k$ or an independent set of size $\\ell$\n(equivalently, the complement graph contains a clique of size $\\ell$).
\n\n We formalize the classical open problem of determining $R(5,5)$, together with the currently best\nknown bounds $43 \\le R(5,5) \\le 46$.
\n\n Note: the diagonal Ramsey number $R(n,n)$ can also be formulated in terms of 2-colorings of\n$2$-subsets, as Combinatorics.hypergraphRamsey 2 n (see FormalConjecturesForMathlib/Combinatorics/Ramsey.lean).
\n
\n [Rad] S. P. Radziszowski,
\n [Exoo89] G. Exoo,
\n [AM24] V. Angeltveit and B. McKay,
\n
\n Two distinct positive integers form an amicable pair if each equals the sum of the\nproper divisors of the other. Equivalently, $(a, b)$ is an amicable pair if\n$\\sigma(a) = a + b$ and $\\sigma(b) = a + b$, where $\\sigma(n)$ denotes the sum of\nall positive divisors of $n$.
\n\n Several open problems about amicable numbers are formalised here:
\n\n Do there exist relatively prime amicable numbers?
\n\n Are there infinitely many amicable pairs?
\n\n Do there exist amicable numbers with opposite parity (one even, one odd)?
\n\n
\n Agrawal's conjecture is a stronger version of the theorem that forms the basis\nof the AKS primality test. If true, it would significantly improve the\nefficiency of primality testing.
\n\n The conjecture states that for coprime $n$ and $r$, if the polynomial congruence\n$(X-1)^n \\equiv X^n-1 \\pmod{n, X^r-1}$ holds, then $n$ is either prime or $n^2 \\equiv 1 \\pmod{r}$.
\n\n
\n A finite group is a Leinster group if the sum of the orders of all its normal subgroups\nequals twice the group's order.
\n\n
\n Leinster, Tom (2001). \"Perfect numbers and groups\".\narXiv:math/0104012
\n\n TODO: The following properties from the Wikipedia article can also be formalized:
\n\n There are no Leinster groups that are symmetric or alternating.
\n\n There is no Leinster group of order p²q² where p, q are primes.
\n\n No finite semi-simple group is Leinster.
\n\n No p-group can be a Leinster group.
\n\n All abelian Leinster groups are cyclic with order equal to a perfect number.
\n\n
\n [Bernard Mathan and Olivier Touli´e,
\n
\n
\n
\n
\n
\n The Modularity conjecture (also know as the Shimura-Taniyama-Weil conjecture) states that\nevery rational elliptic curve is modular, meaning that it can be\nassociated with a modular form. We state the a_p version of the conjecture, which relates the\ncoefficients of the modular form to the number of points on the elliptic curve over finite fields.
\n Since we don't have the conductor of the elliptic curve, our definition of a_p(E) differs from\nthat in the literature at primes of bad reduction. For this reason, we state the conjecture with the\nassumption that p ∤ N, in order to give an equivalent statement.
\n
\n [F. Diamond and J. Shurman,
\n References:
\n\nA2234
\n\n
\n
\n The Pell numbers $P_n$ are defined by $P_0 = 0$,\n$P_1 = 1$, $P_{n+2} = 2*P_{n+1} + P_n$. OEIS A129
\n\n The conjecture says that there are infinitely many prime Pell numbers.
","/FormalConjectures/Wikipedia/Fuglede/":"\n
\n A
\n It is conjectured that every T2, Toronto space is discrete.\nW.R. Brian proved that this holds under GCH.
\n\n
\nThe Toronto problem by
\n The minimum overlap problem asks for the limit of the minimum, over all\nsplittings of ${1, \\ldots, 2n}$ into two sets $A$ and $B$ of equal size, of\nthe maximum number of representations of any integer $k$ as $a - b$ with\n$a \\in A$, $b \\in B$, divided by $n$.
\n\n
\n This file points to the canonical formalization in FormalConjectures.ErdosProblems.«36».
\n
\n Despite extensive empirical evidence—billions of digits have been computed for $\\pi$,\n$e$, and $\\sqrt{2}$, all showing near-uniform digit distribution—it is an open problem\nwhether any of the classical constants $\\pi$, $e$, $\\sqrt{2}$, $\\ln 2$, or $\\varphi$ is\nnormal in any base.
\n\n
\n
\n Artin's conjecture predicts, given an integer $a$, densities of primes $p$ for which\n$a$ is a primitive root modulo $p$. Under certain conditions (when $a$ is not a\npower and its squarefree part is $1\\pmod{4}$) the density is given by Artin's constant\n$$\\prod_{p\\ \\text{prime}} \\left(1 - \\frac{1}{p(p - 1)}\\right).$$\nFor more general values of $a$, this constant must be corrected by certain factors.
\n\n When $a = b^m$, $m$ is a maximal odd power, the squarefree part of $b$ satisfies\n$b_0 \\not\\equiv 1\\pmod{4}$. Then Artin's constant should be multiplied by\n$$\\prod_{p \\mid m} \\frac{p(p - 2)}{p^2 - p - 1}.$$
\n\n When $a = b^m$, $m$ is a maximal power, the squarefree part of $b$ satisfies\n$b_0\\equiv 1\\pmod{4}$. Then Artin's constant should be multiplied by the factor in\nthe above bullet, as well as an additional entanglement factor from the primes dividing\n$\\gcd(b_0, m)$ and primes dividing $b_0$:\n$$1 - \\prod_{p \\mid \\gcd(b_0, m)} \\frac{1}{2 - p}\n\\prod_{p \\mid b_0, p\\nmid m} \\frac{1}{1 + p - p^2}.$$
\n\n When $a = -1$ or $a$ is a square, then the density is $0$.
\n\n Note that Artin's conjecture has been proved subject to the Generalized Riemann Hypothesis\n[Ho67].
\n\n
\nA85397
\n\nLMS14 Lenstra, H.W. et al. \"Character sums for primitive root densities\"
\n [Ho67] Hooley, C. \"On Artin's conjecture.\"
\n
\n Singmaster's conjecture says that for any integer $t>1$, the number of solutions to the equation:
\n\n $\\binom{n}{k} = t,\\quad 1 \\le k < n,$
\n\n with $\\binom{n}{k}$ being the numbers that appear in Pascal's triangle, is bounded by a global\nconstant $O(1)$.
\n\n
\n An integer n : ℤ is (m,k)-perfect if σᵐ(n) = kn where σᵐ is the mᵗʰ iterate of the\nsum of divisors function.
\n
\n
\n
\nmathoverflow/418260\nasked by user Yuan Yang
\n\n D19 in Unsolved Problems in Number Theory\nby
\n
\n
\n A Dedekind number M(n) counts the number of monotone Boolean functions on n variables,\nor equivalently, the number of antichains (Sperner families) in the Boolean lattice 2^[n].
\n For example,\n$$M ( 0 ) = 2 , M ( 1 ) = 3 , M ( 2 ) = 6 , and M ( 3 ) = 20 .$$\nThe first few values grew slowly:\n$$M ( 4 ) = 168 , M ( 5 ) = 7581$$,\nbut then rapidly:\n$$M ( 6 ) = 7828354 , M ( 7 ) = 2414682040998 , M ( 8 ) = 56130437228687557907788$$, and\n$$M ( 9 ) = 286386577668298411128469151667598498812366$$\n(computed in 2023).
\n\n We formalize two definitions:
\n\nM n: the number of monotone Boolean functions (Fin n → Bool) → Bool
\nM' n: the number of antichains (Sperner families) of Finset (Fin n)
\n We prove their values for small n and show that the two definitions agree for all n.
\n The problem is to determine the exact values of $M(n)$ for $n ≥ 10$.\nIn particular, the value of $M(10)$ is currently unknown.
\n\n
\n This file is a Wikipedia-facing entry point for the formalization in\nFormalConjectures.GreensOpenProblems.«72».
\n
\n For every positive natural number $n$, there exists a natural number $m$ with $m ≠ n$, such that\n$φ(n) = φ(m)$ where $φ$ is the Euler totient function.
\n\n
\n [F1998] Kevin Ford. The distribution of totients. https://arxiv.org/abs/1104.3264
\n\n
\n The Hadwiger-Nelson problem asks for the minimum number of colors required to\ncolor the plane such that no two points at unit distance from each other have\nthe same color.
\n\n
\n This file points to the canonical formalization in FormalConjectures.ErdosProblems.«508».
\n
\nRésolution d'une question relative aux déterminants by
\n
\n
\n In this file, we:
\n\n state the conjecture
\n\n state three solved variants of the conjecture, without proof
\n\n prove two solved variants of the conjecture
\n\n prove the conjecture is sharp
\n\n There exists a positive number $C$ such that for any integer $x, y$ with $y^2 \\ne x^3$,\n$|y^2 - x^3| > C \\sqrt{|x|}$.
\n\n
\n L. Danilov,
\n
\n
\n
\n A (base-10)
\n $$a_{0} = n, \\qquad a_{k+1} = a_k + \\operatorname{rev}_{10}(a_k).$$
\n\n One commonly stated conjectural direction is that there are no Lychrel numbers in base 10.\nThe smallest widely studied open case is 196.
\n
\n
\n [Ge92] Gerver, J. L.,
\n [Ro18] Romik, D.
\n [Ba24] Baek, J.
\n
\n
\nWikipedia, Sierpiński number
\n\n [Si60] Sierpiński, W., Elementary Theory of Numbers. Państwowe Wydawnictwo Naukowe,\nWarsaw (1960).
\n\n A positive odd integer $k$ is a
\n The
\n The
\n The
\n A natural number $n$ is called a congruent number if there exists a right triangle with rational\nsides $a$, $b$, and hypotenuse $c$ such that the area of the triangle is $\\frac{1}{2}ab = n$.
\n\n
\n
\n
\n
\n
\n There are two conjectures related to the Ramanujan τ-function:
\n\n Ramanujan-Petersson conjecture: For every prime p, the absolute value of the\nRamanujan τ-function at p is bounded by 2 * p^(11/2).
\n Lehmer's conjecture: The Ramanujan τ-function is never zero for any positive integer n.
\n
\n
\n
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\n
\n
\n The
\n
\nA Survey on the Square Peg Problem\nby
\n
\n
\n A group $G$ is
\n Here equivariance is with respect to the left shift action of $G$ on $A^G$,\ndefined by $(g \\cdot x)(h) = x(g^{-1} h)$, and continuity is with respect to\nthe product topology on $A^G$ (where $A$ carries the discrete topology).
\n\n Gottschalk's conjecture (1973) states that every group is surjunctive.
\n\n
\n Gottschalk, W. H. (1973), \"Some general dynamical notions\"
\n\n
\n
\n A
\n However, a taxicab number is not known for $k=5$, $m=2$, and any $n ≥ 2$:\nNo positive integer is known that can be written as the\nsum of two 5th powers in more than one way, and it is not\nknown whether such a number exists.
\n\n In particular, it is not known whether there exists a\ntaxicab number for $k=5$, $m=2$, and $n=2$.
\n\n
\n
\n
\n An integer n : ℤ can be written as a sum of three cubes (of integers) if and only if\nn is not 4 or 5 mod 9.
\n
\nmathoverflow/100324\nasked by user
\n
\n
\n
\n A perfect number is a positive integer that equals the sum of its proper divisors\n(i.e., all its positive divisors excluding the number itself).
\n\n For example, 6 is perfect because its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6.\nSimilarly, 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28.
\n\n All known perfect numbers are even. Several open problems about perfect numbers are\nformalised here:
\n\n Are there infinitely many perfect numbers?
\n\n Are there infinitely many even perfect numbers?
\n\n Do odd perfect numbers exist?
\n\n
\n
\n
\nReference: https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture
","/FormalConjectures/Wikipedia/Sendov/":"\n
\n Tags: Sendov Conjecture, Ilieff's Conjecture.
","/FormalConjectures/Wikipedia/Lemoine/":"\n
\n [Ki85] Kiltinen, J. and Young P. (1985). Goldbach, Lemoine, and a Know/Don't Know Problem.
\n\n
\nThe fundamental theorem of algebra and complexity theory\nby Steve Smale
\n\n Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$\nthere is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ K* |p'(z)|$ for $K=1$.
\n\n The conjecture has been proven for:
\n\nK = 4\nThe fundamental theorem of algebra and complexity theory\nby
\nK = (d-1)/d if $p$ has real roots or all the roots of $p$ have the same norm.\nCritical points and values of complex polynomials\nby
\n This file is a Wikipedia-facing entry point for the formalization in\nFormalConjectures.ErdosProblems.«20».
\n
\n
\n
\n A sparse ruler of length $L$ is a sequence of marks $0 = a_1 < a_2 < \\dots < a_m = L$.\nA distance $k \\in \\mathbb{N}$ can be measured if there are $i, j \\in {1, \\dots, m}$, such that\n$k = a_j - a_i$.
\n\n One can now ask for rulers that measure every integer up to some $K \\in \\mathbb{N}$ and for them\nto be minimal, i.e. having a minimal number of marks. Furthermore, we can restrict such rulers in\nlength, for example requiring for a ruler of length $L$ to measure every distance up to $L$. This\nis called a perfect ruler and Erdős Problem 170 covers the question of how many marks such minimum\nperfect rulers have asymptotically.
\n\n There are several other questions with regards to sparse rulers and many of them are still unsolved.
","/FormalConjectures/Wikipedia/LonelyRunnerConjecture/":"\n
\n
\n Brocard's problem asks whether the only solutions to $n! + 1 = m^2$ are\n$n = 4, 5, 7$.
\n\n
\n This file points to the canonical formalization in FormalConjectures.ErdosProblems.«398».
\n
\nLeonard Eugene Dickson,
\nArxiv
\n\n
\nLandau Problems Wikipedia Page
\n\nLegendre Conjecture Wikipedia Page
\n\nLuan Alberto Ferreira,
\n An integer $D>0$ is idoneal if every\ninteger that can be expressed in exactly one way (up to order and signs)\nas $x^2 + D y^2$ with gcd(x, Dy)=1 is a prime power or twice a prime power.
\n\n The Idoneal Numbers Completeness Conjecture asserts that the following list of\n65 numbers is complete:\n1,2,3,4,5,6,7,8,9,10,12,13,15,16,18,21,22,24,25,28,30,33,37,40,42,45,48,\n57,58,60,70,72,78,85,88,93,102,105,112,120,130,133,165,168,177,190,210,232,\n240,253,273,280,312,330,345,357,385,408,462,520,760,840,1320,1365,1848.\n
\n
\n [CP96] Chen, H., Gauthier, P. M. \"On Bloch’s constant.\" Journal d’Analyse Mathématique 69 (1996),\n275–291.
\n\n [AG37] Ahlfors, L. V., Grunsky, H. \"Über die Blochsche Konstante.\" Mathematische Zeitschrift 42\n(1937), 671–673.
\n\n [Ya95] Yanagihara, H. \"On the locally univalent Bloch constant.\" Journal d’Analyse Mathématique\n65 (1995), 1–17.
\n\n [Ra43] Rademacher, H. \"On the Bloch-Landau Constant.\"\" American Journal of Mathematics 65 (1943),\n387–390.
\n\n\n [Skin2009] Skinner, Brian. The univalent Bloch constant problem. Complex Variables and Elliptic\nEquations 54 (2009), no. 10, 951–955.
\n\n\n\n
\n
\n
\n [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
\n\n [La10] Lagarias, Jeffrey C., The {$3x+1$} problem: an overview. (2010), 3--29.
\n\n [La16] Lagarias, Jeffrey C., Erdős, Klarner, and the {$3x+1$} problem. Amer. Math. Monthly\n(2016), 753--776.
\n\n [La85] Lagarias, Jeffrey C., The {$3x+1$} problem and its generalizations. Amer. Math. Monthly\n(1985), 3--23.
\n\n
\n
\n
\n
\n The happy ending problem asks whether $f(n) = 2^{n-2} + 1$, where $f(n)$ is the\nsmallest number such that any $f(n)$ points in general position in the plane\ncontain $n$ that form a convex polygon.
\n\n
\n This file points to the canonical formalization in FormalConjectures.ErdosProblems.«107».
\n
\nOn the Gap Conjecture concerning group growth by\n
\n
\n This file adds three conjectures about the Mandelbrot and Multibrot sets:
\n\n the
\n the
\n the conjecture that the boundaries of these sets have zero area.\nThe first two conjectures are related in that the former implies the latter.
\n\n
\n
\n
\n The actual formalization is in FormalConjectures.HilbertProblems.«5».
\n Hilbert's fifth problem asks whether every locally Euclidean topological group admits a Lie group\nstructure. This was resolved affirmatively by Gleason, Montgomery, and Zippin in 1952.
\n\n
\n
\n
\n
\n [PPVW2016] Jennifer Park, Bjorn Poonen, John Voight, and Melanie Matchett Wood.\nA heuristic for boundedness of ranks of elliptic curves,\nhttps://ems.press/journals/jems/articles/16228
\n\n [BS2013] Manjul Bhargava and Arul Shankar. The average size of the 5-Selmer group of\nelliptic curves is 6, and the average rank is less than 1, https://arxiv.org/pdf/1312.7859
\n\n\n
\n
\nWikipedia: Examples of numerical series
\n\nMathWorld: Flint Hills Series
\n\n\n\n
\nWikipedia - Catalan's conjecture
\n\narXiv:2507.12397 (Lebesgue-Nagell equation)
\n\n This file points to the canonical formalization in FormalConjectures.ErdosProblems.«274».
\n
\n We define the notion of regular primes, which are prime numbers that are coprime with the\ncardinality of the class group of the p-th cyclotomic field. We also state that there are\ninfinitely many regular primes.
\n
\n
\n This file contains a number of open problems related to the minimal size of a square (or circle)\nthat can contain a given number of unit squares (or circles).\nIn each case, we provide a known upper bound, and ask for the least such size.
\n\n
\nWikipedia on packing of squares
\n\nWikipedia on packing of circles in a circle
\n\nWikipedia on packing of circles in a square
\n\n Friedman, Erich (2009), \"Packing unit squares in squares: a survey and new results\",\nElectronic Journal of Combinatorics, 1000, Dynamic Survey 7
\n\n A website with visualizations of packings:\nlink
\n\n
\n [Sh12] Shapirov, Ruslan. Perfect cuboids and irreducible polynomials. https://arxiv.org/abs/1108.5348
\n\n
\n by B. Curtin, G. R. Pourgholi
\n\n
\n by L. Pyber\n
\n
\nTao's optimization constant 1a
\n\n [M2010] Matolcsi, Máté, and Carlos Vinuesa. \"Improved bounds on the supremum of autoconvolutions.\"\nJournal of mathematical analysis and applications 372.2 (2010): 439-447. arXiv:0907.1379
\n\n [Y2026] Yuksekgonul, Mert et al., \"Learning to Discover at Test Time,\" 2026, arXiv:2601.16175
\n\n More commonly known as the no-three-in-line problem.
\n\n Given $N \\lt 2$ and a more than $2 * N$ points on an $N \\times N$-grid,\nare there $3$ of the points on a common line?
\n\n
\n [GK2025] Grebennikov, A. Kwan, M. No $(k + 1)$-in-line problem for large constant $k$.\nhttps://arxiv.org/abs/2510.17743
\n\n Does Ulam's sequence have positive density?\nCan one explain the curious Fourier properties of Ulam's sequence?
\n\n
\n Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$.\nIs every $x \\in {1, \\ldots, p-1}$ congruent to some product $a_1 a_2$ where $a_1, a_2 \\in A$?
\n\n This is a problem of Erdős, Odlyzko, and Sárközy [105] from 1987.
\n\n
\n
\n [La79] Lovász, László. \"On the Shannon capacity of a graph.\"\nIEEE Transactions on Information theory 25.1 (1979): 1-7.
\n\n [Po20] Polak, Sven. \"New methods in coding theory: Error-correcting codes and the Shannon capacity.\"\narXiv preprint arXiv:2005.02945 (2020).
\n\n
\n References:
\n\n [Gr24] Green, Ben. \"100 open problems.\" (2024).
\n\n [CCW99] Carbery, Anthony, Michael Christ, and James Wright. \"Multidimensional van der Corput and sublevel set estimates.\" Journal of the American Mathematical Society 12.4 (1999): 981-1015 Section 6.
\n\n [Ke00] Keleti, Tamás. \"Density and covering properties of intervals of ℝn.\" Mathematika 47.1-2 (2000): 229-242.
\n\n [KKM02] Katz, Nets Hawk, Elliot Krop, and Mauro Maggioni. \"Remarks on the box problem.\" Mathematical Research Letters 9.4 (2002): 515-520.
\n\n [Mu02] Mubayi, Dhruv. \"Some exact results and new asymptotics for hypergraph Turán numbers.\" Combinatorics, Probability and Computing 11.3 (2002): 299-309 Conjecture 1.4.
\n\n [CPZ20] Conlon, David, Cosmin Pohoata, and Dmitriy Zakharov. \"Random multilinear maps and the Erd\\H {o} s box problem.\" arXiv preprint arXiv:2011.09024 (2020).
\n\n What is the smallest subset of ℕ containing, for each d = 1, …, N,\nan arithmetic progression of length k with common difference d?
\n
\nGreen & Tao,
\n Write $F(N)$ for the largest Sidon subset of $[N]$.\nImprove, at least for infinitely many $N$, the bounds $N^{1/2} + O(1) \\le F(N) \\le N^{1/2} + N^{1/4} + O(1)$.
\n\n Note: the upper bound was improved to $N^{1/2} + 0.98183 N^{1/4} + O(1)$ in [CHO25].
\n\n Related to Erdős Problem 30.
\n\n
\n [Gr24] [Ben Green's Open Problem 31](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.7 Problem 31)
\n\n [Gr01] Green, Ben. \"The number of squares and $ B_h [g] $ sets.\"\nActa Arithmetica 100.4 (2001): 365-390.
\n\n [BFR23] Balogh, József, Zoltán Füredi, and Souktik Roy. \"An upper bound on the size of Sidon sets.\"\nThe American Mathematical Monthly 130.5 (2023): 437-445.
\n\n [CHO25] Carter, Daniel, Zach Hunter, and Kevin O’Bryant. \"On the diameter of finite Sidon sets.\"\nActa Mathematica Hungarica 175.1 (2025): 108-126.
\n\n [ET41] Erdos, Paul, and Pál Turán. \"On a problem of Sidon in additive number theory, and on some\nrelated problems.\" J. London Math. Soc 16.4 (1941): 212-215.
\n\n [Li69] Lindström, Bernt. “A remark on B4-Sequences.” Journal of Combinatorial Theory,\nSeries A 7 (1969): 276-277.
\n\n [CLZ01] Cohen, G.D., Litsyn, S., & Zémor, G. (2001). Binary B2-Sequences : A New Upper Bound.\nJ. Comb. Theory A, 94, 152-155.
\n\n References:
\n\n [Gr24] Green, Ben. \"100 open problems.\" (2024).
\n\n [BJP14] T. Brown, V. Jungić and A. Poelstra, \"On double 3-term arithmetic progressions\",\nIntegers 14 (2014), Paper No. A43.
\n\n [CCS14] J. Cassaigne, J. D. Currie, L. Schaeffer and J. Shallit, \"Avoidance of additive cubes and\nrelated results\", Adv. in Appl. Math. 56 (2014), 25–66.
\n\n References:
\n\n [Gr24] Green, Ben. \"100 open problems.\" (2024).
\n\n [BlSi20] Bloom, Thomas F., and Olof Sisask. \"Breaking the logarithmic barrier in Roth's theorem on\narithmetic progressions.\" arXiv preprint arXiv:2007.03528 (2020).
\n\n References:
\n\n [Gr24] Green, Ben. \"100 open problems.\" (2024).
\n\n [JLP92] Jaeger, François, et al. \"Group connectivity of graphs—a nonhomogeneous analogue of\nnowhere-zero flow properties.\" Journal of Combinatorial Theory, Series B 56.2 (1992): 165-182.
\n\n [ALM91] Alon, Noga, Nathan Linial, and Roy Meshulam. \"Additive bases of vector spaces over prime\nfields.\" Journal of Combinatorial Theory, Series A 57.2 (1991): 203-210.
\n\n [Yu25] Yu, Yang. \"Note on the Additive Basis Conjecture.\" arXiv preprint arXiv:2510.01300 (2025).
\n\n
\n Let $G$ be a finite abelian group. Consider the space $\\Phi(G)$ of all functions on $G$ which\nare \"convex combinations\" (in the sense of complex coefficients $c_i$ with\n$\\sum |c_i| \\le 1$) of functions of the form\n$$\\phi(g) := \\mathbb{E}
\n Let $\\Phi'(G)$ be the space defined similarly, but with $f_3(x_1, x_2)$ required to be\na function of $x_1 + x_2$. Do $\\Phi(G)$ and $\\Phi'(G)$ coincide?
\n\nNote: The \"convex combination\" here uses complex coefficients whose absolute values sum to\nat most 1 (cf. personal communication with B. Green, April 2026). Since the base sets are\nbalanced (closed under multiplication by unit complex numbers), this absolutely convex hull\nequals the real convex hull of the complex-valued base set.
\n\nMotivation: $\\Phi(G)$ is a 'generalised convolution algebra' as considered by\nConlon–Fox–Zhao, whereas $\\Phi'(G)$ consists of Tao's $\\text{UAP}_2(G)$-functions.
","/FormalConjectures/GreensOpenProblems/«77»/":"\n
\n
\n
\n
\n [Gr24] Green, Ben. \"100 open problems.\" (2024).
\n\n [AKS14] Ahmed, Tanbir, Oliver Kullmann, and Hunter Snevily. \"On the van der Waerden numbers\nw (2; 3, t).\" Discrete Applied Mathematics 174 (2014): 27-51.
\n\n [KeMe23] Kelley, Zander, and Raghu Meka. \"Strong bounds for 3-progressions.\" 2023 IEEE 64th\nAnnual Symposium on Foundations of Computer Science (FOCS). IEEE, 2023.
\n\n [Hu22] Hunter, Zach. \"Improved lower bounds for van der Waerden numbers.\" Combinatorica 42.\nSuppl 2 (2022): 1231-1252.
\n\n [Gr21] Green, Ben. \"New lower bounds for van der Waerden numbers.\" Forum of Mathematics,\nPi. Vol. 10. Cambridge University Press, 2022.
\n\n [Sc20] Schoen, Tomasz. \"A subexponential upper bound for van der Waerden numbers W (3, k).\"\narXiv preprint arXiv:2006.02877 (2020).
\n\n [BLR08] Brown, Tom, Bruce M. Landman, and Aaron Robertson. \"Bounds on some van der Waerden\nnumbers.\" Journal of Combinatorial Theory, Series A 115.7 (2008): 1304-1309.
\n\n [LiSh10] Li, Yusheng, and Jinlong Shu. \"A lower bound for off-diagonal van der Waerden numbers.\"\nAdvances in Applied Mathematics 44.3 (2010): 243-247.
\n\n
\n [Gr24] Ben Green's Open Problem 33
\n\n [CaHa20] Caprace, Pierre-Emmanuel, and Pierre de la Harpe. \"Groups with irreducibly unfaithful\nsubsets for unitary representations.\" Confluentes Mathematici 12.1 (2020): 31-68.
\n\n [CrLe07] Croot, Ernie, and Vsevolod F. Lev. \"Open problems in additive combinatorics.\"\nAdditive combinatorics 43.207-233 (2007): 1.
\n\n
\n [Gr24] Green, Ben. \"100 open problems.\" (2024).
\n\n [BJR11] Bollobás, Béla, Svante Janson, and Oliver Riordan. \"On covering by translates of a set.\"\nRandom Structures & Algorithms 38.1‐2 (2011): 33-67.
\n\n
\n
\n
\n [Gr24] Ben Green's Open Problem 41
\n\n [Ma15] Manners, Freddie. \"A solution to the pyjama problem.\" Inventiones mathematicae 202.1 (2015): 239-270.
\n\n [KrLe25] Kravitz, Noah, and James Leng. \"Quantitative pyjama.\" arXiv preprint arXiv:2510.17744 (2025).
\n\n Let $A$ be the smallest set containing $2$ and $3$ and such that $a_1a_2 - 1 \\in A$\nif $a_1, a_2 \\in A$. Does $A$ have positive density?
\n\n
\n This file points to the canonical formalization in FormalConjectures.ErdosProblems.«424».
\n
\n [Gr26] Ben Green's Open Problems
\n\n [FSS20] Fox, Jacob, et al. \"Triforce and corners.\" Mathematical Proceedings of the Cambridge\nPhilosophical Society. Vol. 169. No. 1. Cambridge University Press, 2020.
\n\n [Ma21] Mandache, Matei. \"A variant of the Corners theorem.\" Mathematical Proceedings of the\nCambridge Philosophical Society. Vol. 171. No. 3. Cambridge University Press, 2021.
\n\n [Ch11] Chu, Qing. \"Multiple recurrence for two commuting transformations.\" Ergodic Theory and\nDynamical Systems 31.3 (2011): 771-792.
\n\n References:
\n\n [Gr24] Green, Ben. \"100 open problems.\" (2024).
\n\n [Be23] Bedert, Benjamin. \"On unique sums in Abelian groups.\" Combinatorica 44.2 (2024): 269-298.
\n\n [St76] Straus, E. G. \"Differences of residues (mod p).\" Journal of Number Theory 8.1 (1976): 40-42.
\n\n Suppose that $A \\subset \\mathbb{F}_2^n$ is a set of density $\\alpha$. Does $10A$ contain a coset\nof some subspace of dimension at least $n - O(\\log(1/\\alpha))$?
\n\n Here $kA$ denotes the $k$-fold iterated sumset, i.e., the set of all sums of $k$ elements from $A$\n(with repetition allowed). In Mathlib, this is denoted k • A using pointwise scalar\nmultiplication on sets.
\n
\n
\n [Gr26] Ben Green's Open Problem 18
\n\n [Au16] Austin, Tim. \"Ajtai–Szemerédi theorems over quasirandom groups.\" Recent trends in\ncombinatorics. Cham: Springer International Publishing, 2016. 453-484.
\n\n [So13] Solymosi, Jozsef. \"Roth-type theorems in finite groups.\" European Journal of Combinatorics\n34.8 (2013): 1454-1458.
\n\n [Go01] Gowers, William T. \"A new proof of Szemerédi's theorem.\" Geometric & Functional Analysis\nGAFA 11.3 (2001): 465-588.
\n\n Can we pick residue classes $a_p \\pmod{p}$, one for each prime $p \\leq N$,\nsuch that every integer $\\leq N$ lies in at least 10 of them?
\n\n
\n This file points to the canonical formalization in FormalConjectures.ErdosProblems.«689».
\n Estimate the infimum of the $L^p$ norm of the self-convolution of a nonnegative integrable\nfunction supported on $[0,1]$ with total integral $1$.
\n\n We model a function f : [0,1] → ℝ≥0 as a function f : ℝ → ℝ that is nonnegative, integrable,\nsupported on [0,1], and has total integral 1.
\n
\nGr01\nB. J. Green,
\nCS17\nA. Cloninger and S. Steinerberger,
\nMV10\nM. Matolcsi and C. Vinuesa,
\n
\n
\n [Gr24] Ben Green's Open Problem 40
\n\n [Da90] Davydov, Alexander Abramovich. \"Construction of linear covering codes.\"\nProblemy Peredachi Informatsii 26.4 (1990): 38-55.
\n\n [CHL97] Cohen, G., Honkala, I., Litsyn, S., & Lobstein, A. (1997). Covering codes (Vol. 54). Elsevier.
\n\n [St94] R. Struik, Covering codes, PhD Thesis, Eindhoven University of Technology, the Netherlands, 106 pp, 1994.
\n\n
\n This problem was originally considered by Erdős and Newman.
","/FormalConjectures/GreensOpenProblems/«81»/":"\n Let $A$ be a set of size $n$ integers. Is there some absolute constant $c > 0$ and $\\theta$\nsuch that $\\sum_{a \\in A} \\cos(a \\theta) \\leq - c \\sqrt{n}$?
\n\n
\n This file points to the canonical formalization in FormalConjectures.ErdosProblems.«510».
\n
\n References:
\n\n
\n [Gr24] Green's Open Problems #36
\n\n [CKS05] Cohn, H., Kleinberg, R., Szegedy, B., and Umans, C. \"Group-theoretic Algorithms for\nMatrix Multiplication\" (Problem 4.7)
\n\n References:
\n\nGreen, Ben. \"100 open problems.\" (2024).
\n\n [FrKlMo25] Frantzikinakis, N., O. Klurman, and J. Moreira. \"Partition regularity of Pythagorean pairs.\" Forum of Mathematics, Pi 13. Cambridge University Press (2025).
\n\n References:
\n\nGreen, Ben. \"100 open problems.\" (2024).
\n\n [Aa19] Aaronson, James. \"Maximising the number of solutions to a linear equation in a set of integers.\"\nBulletin of the London Mathematical Society 51.4 (2019): 577-594.
\n\n [HaL28] Hardy, G. H., and J. E. Littlewood. \"Notes on the theory of series (VIII): an inequality.\"\nJournal of the London Mathematical Society 1.2 (1928): 105-110.
\n\n
\n [Gr26] Ben Green's Open Problems
\n\n [Mo17] Moreira, Joel. \"Monochromatic sums and products in N.\" Annals of Mathematics 185.3 (2017):\n1069-1090.
\n\n [GrSa25] Green, Ben, and Mehtaab Sawhney. \"Bounds for monochromatic solutions to\n${x+ y, xy} $.\" arXiv preprint arXiv:2511.09365 (2025).
\n\n [Ri25] Richter, Florian K. \"Sums and products in sets of positive density.\" arXiv preprint\narXiv:2507.00515 (2025).
\n\n [BoSa24] Bowen, Matt, and Marcin Sabok. \"Monochromatic products and sums in the rationals.\" Forum\nof Mathematics, Pi. Vol. 12. Cambridge University Press, 2024.
\n\n [Bo25] Bowen, Matt. \"Monochromatic products and sums in 2-colorings of N.\" Advances in Mathematics\n462 (2025): 110095.
\n\n [Al23] Alweiss, Ryan. \"Monochromatic Sums and Products over $\\mathbb {Q} $.\" arXiv preprint\narXiv:2307.08901 (2023).
\n\n
\n [Gr12] Green, Ben. \"What is... an approximate group.\" Notices Amer. Math. Soc 59.5 (2012): 655-656.
\n\n [Br13] Breuillard, Emmanuel, Ben Green, and Terence Tao. \"Small doubling in groups.\"\nErdős Centennial. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. 129-151.
\n\n [Sa10] Sanders, Tom. \"On a nonabelian Balog–Szemerédi-type lemma.\" Journal of the Australian\nMathematical Society 89.1 (2010): 127-132.
\n\n [CrSi10] Croot, Ernie, and Olof Sisask. \"A probabilistic technique for finding almost-periods of\nconvolutions.\" Geometric and functional analysis 20.6 (2010): 1367-1396.
\n\n
\n [Ruzsa](I. Z. Ruzsa, Solving a linear equation in a set of integers. I. Acta Arith. 65 (1993), no. 3, 259–282.)
\n\n [Schoen and Sisask](T. Schoen and O. Sisask, Roth’s theorem for four variables and additive structures in sums of sparse sets Forum of Mathematics, Sigma (2016), Vol. 4, e5, 28 pages.)
\n\n [Yufei Zhao](Via Personal Communication with Ben Green)
\n\n
\n [Gr24] Green, Ben. \"100 open problems.\" (2024).
\n\n [Sh20] Shakan, George. \"A Large Gap in a Dilate of a Set.\" SIAM Journal on Discrete Mathematics\n34.4 (2020): 2553-2555.
\n\n
\n Original formulation: M. Talagrand,
\n References:
\n\n [Gr24] Green, Ben. \"100 open problems.\" (2024).
\n\n [ESS89] Erdős, Pál, András Sárközy, and V. T. Sós. \"On a conjecture of Roth and some related\nproblems I.\" Irregularities of partitions. Berlin, Heidelberg: Springer Berlin Heidelberg, 1989.\n47-59.
\n\n [Ru04] Ruzsa, Imre Z. \"A problem on restricted sumsets.\" CONTEMPORARY MATHEMATICS 342 (2004):\n245-248.
\n\n References:
\n\nGreen, Ben. \"100 open problems.\" (2024).
\n\nMathoverflow/339137 asked by user Sil
\n\nMathStackexchange/3325163 asked by user Emmanuel Amiot
\n\n References:
\n\n [Gr24] Green, Ben. \"100 open problems.\" (2024).
\n\n [Er65] P. Erdős. Extremal problems in number theory, In Proc. Sympos. Pure Math., Vol. VIII,\npages 181–189. Amer. Math. Soc., Providence, R.I., 1965.
\n\n [Sa21] Sanders, Tom. \"The Erdős–Moser Sum-free Set Problem.\" Canadian Journal of Mathematics 73.1\n(2021): 63-107.
\n\n [Ru05] I. Z. Ruzsa, Sum-avoiding subsets. Ramanujan J., 9 (2005) (1-2):77–82.
\n\n [Ch71] S. L. G. Choi. On a combinatorial problem in number theory. Proc. London Math. Soc. (3),\n23:629–642, 1971. doi:10.1112/plms/s3-23.4.629.
\n\n [BSS00] A. Baltz, T. Schoen, and A. Srivastav. Probabilistic construction of small strongly\nsum-free sets via large Sidon sets. Colloq. Math., 86(2):171–176, 2000.\ndoi:10.4064/cm-86-2-171-176.
\n\n
\n [Gr24] Ben Green's 100 Open Problems
\n\n [Er80] Erdős, Paul. \"A survey of problems in combinatorial number theory.\"\nAnnals of Discrete Mathematics 6 (1980): 89-115.
\n\n A conjecture of Lam and Litt on algebraic solutions of algebraic ODEs.
\n\n Let $g \\in \\mathbb{Q}(z, y_0, \\dots, y_{n-1})$ be a rational function in\n$n + 1$ variables. Let $f$ be a power series over $\\mathbb{Q}$ such that\n$f^{(n)}(z) = g(z, f(z), f'(z), \\dots, f^{(n-1)}(z))$.\nAlso, assume that $g(0, f(0), f'(0), \\dots, f^{(n-1)}(0))$ is defined.\nThen the following are equivalent:
\n\n $f$ is algebraic over $\\mathbb{Q}[z]$.
\n\n There exists $N$ such that for all $n$, the $n$-th coefficient of $f$ is in $\\mathbb{Z}[1/N]$.
\n\n There exists an integer-valued function $\\omega$ on the set of primes with\n$\\lim_{p \\to \\infty} \\omega(p) / p = \\infty$ such that, for each prime $p$,\nthe rational numbers $a_0, a_1, \\dots, a_{\\omega(p)}$ are in $\\mathbb{Z}_{(p)}$.
\n\n The implication 1) => 2) is due to Eisenstein, and 2) => 3) is trivial.
\n\n
\n Yeuk Hay Joshua Lam, Daniel Litt, \"Algebraicity and integrality of solutions to differential equations\",\narxiv/2501.13175
\n\n Gotthold Eisenstein. \"Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Funktionen\",\nBericht der Königl. Preuss. Akademie der Wissenschaften zu Berlin, 1852
\n\n TODO:
\n\n Lam-Litt conjecture implies Grothendieck p-curvature conjecture.
\n\n Examples in Remark 1.1.3 and 1.1.5 on the conditions of the conjecture.
\n\n The Beaver Math Olympiad (BMO) is a set of mathematical reformulations of the halting/nonhalting\nproblem of specific Turing machines from all-0 tape. These problems came from studying small Busy\nBeaver values. Some problems are open and have a conjectured answer, some are open and don't have a\nconjectured answer, and, some are solved.
\n\n Among these problems is the Collatz-like
\n For some BMO problem, the equivalence between the mathematical formulation and the corresponding\nTuring machine non-termination has been formally proved in Rocq, we indicate it when done.
\n\n
\n In the literature it is known that every convex set in ℝ² has VC dimension at most 3,\nand there exists a convex set in ℝ³ with infinite VC dimension (even more strongly,\nwhich shatters an infinite set).
\n\n This file states that every convex set in ℝⁿ has finite VCₙ dimension, constructs a convex set in\nℝⁿ⁺² with infinite VCₙ dimension (even more strongly, which n-shatters an infinite set),\nand conjectures that every convex set in ℝⁿ⁺¹ has finite VCₙ dimension.
","/FormalConjectures/Other/SchurTruncatedExponential/":"\n
\n
\n
\n Let $A$ and $B$ be sets of words of length $n$ over an alphabet with $q$ letters. If no\n(nonempty) suffix of any word in $A$ coincides with a prefix of any word in $B$, then\n$$|A| \\cdot |B| \\leq \\frac{q^{2n}}{en}.$$
\n\n
\n [Maximal sets of strings with no prefix-suffix overlap]\n(https://mathoverflow.net/questions/508648/maximal-sets-of-strings-with-no-prefix-suffix-overlap)\nby
\nAn isoperimetric inequality for word overlap\nby
\nProblem: For which numbers of parties $n$ and local dimensions $d$ does there\nexist a pure absolutely maximally entangled state $\\psi$?
\n\n A pure state $\\psi$ on $n$ parties of local dimension $d$ is called\nabsolutely maximally entangled (AME) if, for every subset of at most half\nof the parties, the corresponding reduced density matrix is maximally mixed.
\n\n
\n Open Quantum Problems, Problem 35:\nhttps://oqp.iqoqi.oeaw.ac.at/existence-of-absolutely-maximally-entangled-pure-states
\n\n Formal Conjectures issue #3452:\nhttps://github.com/google-deepmind/formal-conjectures/issues/3452
\n\n W. Helwig, W. Cui, A. Riera, J. I. Latorre, and H.-K. Lo,\n
\n D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, and K. Życzkowski,\n
\n A. Higuchi and A. Sudbery,\n
\n A. J. Scott,\n
\n F. Huber, O. Gühne, and J. Siewert,\n
\n F. Huber and M. Grassl,\n
\n S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioć,\nA. Lakshminarayan, and K. Życzkowski,\n
\n G. Rajchel-Mieldzioć, R. Bistroń, A. Rico, A. Lakshminarayan,\nand K. Życzkowski,\n
\n This file formalizes the problem of determining for which pairs $(n,d)$ there exists an\nabsolutely maximally entangled pure state $\\mathrm{AME}(n,d)$.
\n\n We represent an $n$-partite state of local dimension $d$ by the finite-dimensional Hilbert space\nEuclideanSpace ℂ (Config n d), whose coordinates in the computational basis are amplitudes.\nThe helper mkStateVector turns an amplitude function into such a state, and normalization is\nimposed explicitly via IsNormalized, i.e. via the ambient $L^2$ norm.
\n The main reusable lemma is reducedDensityFirst_of_completion: if a state is a\nuniform superposition over the graph of an injective completion function\ncompletion : Config m d → Config (n - m) d,\nthen the reduced state on the first $m$ parties is maximally mixed.
\n As demonstration, we show that the Bell states with $n=2$ and GHZ states with $n=3$ are\nAME states, and the GHZ state with $n=4$ is not an AME state.
","/FormalConjectures/OpenQuantumProblems/«23»/":"\n The OQP page presents three increasingly strong formulations of this problem.\nIn this file we formalize the first one, closest to the physics terminology:\nexistence of a symmetric informationally complete POVM in every finite dimension.
\n\n A SIC-POVM in dimension $d$ can be represented by a family of $d^2$ normalized\nvectors in $\\mathbb{C}^d$ whose pairwise squared overlaps are all equal to\n$(d + 1)^{-1}$. We encode such a family as a map Fin (d ^ 2) → StateVector d.
\n SIC-POVMs are a basic structure in finite-dimensional quantum information.\nThey are closely related to equiangular lines, tight frames, quantum state\nreconstruction, and finite-dimensional measurement theory.\nThe open problem asks whether such families exist in every dimension.
\n\n This file formalizes the existence problem for symmetric informationally complete\nPOVMs through the predicate HasSICPOVM d.
\n More precisely, it contains the following layers.
\n\n The main definitions formalized in this file are:
\n\nStateVector d: a state vector in ℂ^d;
\nmkStateVector: constructor from coordinates in the computational basis;
\nIsNormalized ψ: normalization predicate for a state vector;
\noverlapSq φ ψ: squared magnitude of the inner-product overlap;
\nHasConstantOverlapSq c Φ: constant pairwise squared-overlap condition;
\nsicOverlapSq d: the SIC overlap value (d + 1)⁻¹;
\nIsSICFamily d Φ: the predicate that a family of d^2 vectors in ℂ^d\nis a SIC family;
\nHasSICPOVM d: existence of a SIC family in dimension d.
\n In addition, the file includes explicit witness families and convenient\nconstructors used in the low-dimensional benchmark cases:
\n\nvec2, vec3;
\nqubitSICFamily;
\nhesseFamily;
\nbb84Family.
\n The main open theorem is:
\n\nsicPOVMs, expressing the conjecture that for every d ≥ 1, there exists a\nSIC-POVM in dimension d.
\n The file also isolates several special cases:
\n\n solved low-dimensional benchmark cases:\nhasSICPOVM_zero, hasSICPOVM_one, hasSICPOVM_two, hasSICPOVM_three;
\n a negative benchmark result:\nbb84Family_not_isSICFamily, showing that the BB84 family in dimension 2\ndoes not form a SIC family;
\n selected open benchmark dimensions:\nhasSICPOVM_56, hasSICPOVM_58, hasSICPOVM_59, hasSICPOVM_60,\nhasSICPOVM_64, hasSICPOVM_68, hasSICPOVM_69, hasSICPOVM_70,\nhasSICPOVM_71, hasSICPOVM_72, hasSICPOVM_75.
\n The file includes the following test lemmas and benchmark-support statements:
\n\nhasConstantOverlapSq_singleton;
\nsicOverlapSq_one, sicOverlapSq_two, sicOverlapSq_three,\nsicOverlapSq_pos;
\nisSICFamily_singleton_iff, isSICFamily_one_of_normalized;
\nqubitSICFamily_normalized, qubitSICFamily_pairwise;
\nhesseFamily_normalized, hesseFamily_pairwise;
\nbb84Family_normalized.
\n At present, these @[category test, AMS 15 47 81] results are included with\nplaceholder proofs by sorry; they are intended to be proved in the next PR.
\n
\n IQOQI Vienna Open Quantum Problems, problem 23:\nhttps://oqp.iqoqi.oeaw.ac.at/sic-povms-and-zauners-conjecture
\n\n Formal Conjectures issue #1823:\nhttps://github.com/google-deepmind/formal-conjectures/issues/1823
\n\n J. M. Renes, R. Blume-Kohout, A. J. Scott, and M. C. Caves,\n
\n G. Zauner,\n
\n For each integer $d \\ge 2$, determine the maximum number $k$ for which there exist\northonormal bases $\\mathcal{B}_1, \\dots, \\mathcal{B}_k$ of the complex Hilbert space\n$\\mathbb{C}^d$ such that any two distinct bases are mutually unbiased.
\n\n Concretely, if\n$\\mathcal{B}
\n The problem is therefore to determine the maximal value\n$\\mu(d) := \\max { k : \\text{there exist } k \\text{ pairwise mutually unbiased\northonormal bases in } \\mathbb{C}^d }$.
\n\n In this file, an orthonormal basis is represented by a unitary matrix whose columns are the\nbasis vectors. For two such bases U and V, the matrix relativeUnitary U V, which is\n$U^\\dagger V$, contains all cross-basis overlaps as its entries. Since Lean works more\nsmoothly with squared norms, we formalize mutual unbiasedness by requiring\n$| (relativeUnitary\\ U\\ V)_{ij} |^2 = 1 / d$\nfor all $i, j$, which is equivalent to\n$|\\langle e_i^{(r)}, e_j^{(s)} \\rangle| = d^{-1/2}$.
\n Mutually unbiased bases are a basic structure in finite-dimensional quantum theory.\nThey arise in quantum state determination, quantum tomography, quantum cryptography,\nfinite geometry, and combinatorics.
\n\n A general upper bound is $\\mu(d) \\le d + 1$.\nEquality is known when $d$ is a prime power, via constructions over finite fields.\nFor composite dimensions that are not prime powers, the exact value of $\\mu(d)$ is in\ngeneral open.
\n\n The smallest and most famous unresolved case is $d = 6$.\nThe IQOQI OQP page emphasizes this dimension in particular: although many equivalent\nreformulations are known, no construction yielding more than three mutually unbiased bases\nin dimension six is known.
\n\n This file is organized around the quantity IsMaxMUBCount d k, which expresses that\n$k$ is the maximum number of mutually unbiased orthonormal bases in dimension $d$.
\n the open theorem mutuallyUnbiasedBases expresses the full problem for all $d \\ge 2$;
\n the open theorem mutuallyUnbiasedBases_dim6 expresses the especially important case\n$d = 6$;
\n the solved theorem mutuallyUnbiasedBases_dim2 proves the qubit case $\\mu(2) = 3$.
\n
\n IQOQI Vienna Open Quantum Problems, problem 13:\nhttps://oqp.iqoqi.oeaw.ac.at/mutually-unbiased-bases
\n\n Master list of open quantum problems:\nhttps://oqp.iqoqi.oeaw.ac.at/open-quantum-problems
\n\n I. D. Ivanović,\n
\n W. K. Wootters and B. D. Fields,\n
\n A. Klappenecker and M. Rötteler,\n
\n M. Grassl,\n
\n P. Butterley and W. Hall,\n
\n S. Brierley and S. Weigert,\n
\n P. Raynal, X. Lü, and B.-G. Englert,\n
\n The dimension-six case is not known to be solved. At present, the best-known general picture is:
\n\n $3 \\le \\mu(6) \\le 7$,
\n\n complete sets of $7$ MUBs are not known,
\n\n and several analytic and numerical works give strong evidence that one cannot go beyond $3$.
\n\n This is why the theorem mutuallyUnbiasedBases_dim6 is marked as an open research statement.
\n
\n
\n This file contains tests for graph invariants on 5 specific concrete graphs:
\n\nHouseGraph: A graph on 5 vertices.
\nK4: The complete graph on 4 vertices.
\nPetersenGraph: The Petersen graph on 10 vertices.
\nC6: The cycle graph on 6 vertices.
\nStar5: The star graph with 5 leaves (6 vertices total).
\n Tests cover:\nindependence_number, dominationNumber, average_distance, diameter, radius,\ngirth, order, size, szeged_index, wiener_index, min_degree, max_degree,\naverage_degree, matching_number, residue, annihilation_number, cvetkovic.
","/FormalConjectures/WrittenOnTheWallII/GraphConjecture19/":"\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n This file studies the existence of
\n every non-monochromatic inherited vertex colouring has total weight 0, while
\n each of the D monochromatic colourings has total weight 1.
\n In the quantum-optics motivation, such a construction corresponds to generating high-dimensional\nmultipartite GHZ-type states using probabilistic pair sources and linear optics (without additional\nresources), where interference patterns can be expressed as weighted sums over perfect matchings.
\n\n For N = 4 and D ≥ 4, does there exist such a graph/weighting?
\n For even N ≥ 6 and D ≥ 3, does there exist such a graph/weighting?
\n A quantum graph with N vertices and D colours can be encoded by a weight function\nW : EdgeN N D α → α (for a coefficient domain α).
\n For each assignment of vertex indices ι : V N → Fin D, we define a perfect-matching sum\npmSumN N D W ι (a sum over perfect matchings, where each matching contributes the product of the\ncorresponding edge weights determined by ι). The equation system EqSystemN N D W requires
\npmSumN N D W ι = 1 iff ι is constant (all entries equal), and 0 otherwise.
\n The open conjectures in this file ask for non-existence/existence of such W over various\ncoefficient domains (e.g. ℂ, ℝ, ℤ, and restricted integer weights).
\n [Krenn2017] M. Krenn, X. Gu, A. Zeilinger,\n\"Quantum Experiments and Graphs: Multiparty States as Coherent Superpositions of Perfect Matchings\",\n
\n [MO2018] Vertex coloring inherited from perfect matchings (motivated by quantum physics),\nMathOverflow question 311325.
\n\n [Gu2019] X. Gu, M. Erhard, A. Zeilinger, M. Krenn,\n\"Quantum experiments and graphs II: Quantum interference, computation, and state generation\",\n
\n [Krenn2019] Questions on the Structure of Perfect Matchings inspired by Quantum Physics\nby
\n [Chandran2022] Edge-coloured graphs with only monochromatic perfect matchings and their connection to quantum physics\nby
\n [Chandran2024] Krenn–Gu conjecture for sparse graphs\nby
\n
\n Fix m ≥ 2. Consider the directed graph with vertex set (ZMod m)³, where from each vertex\n(i, j, k) there are directed arcs to (i+1, j, k), (i, j+1, k), and (i, j, k+1)\n(arithmetic mod m). The goal is to partition all 3m³ directed arcs into three\nedge-disjoint directed Hamiltonian cycles (each of length m³).
\n Knuth describes an explicit construction, found by Claude (Anthropic), that achieves this\ndecomposition for all odd m ≥ 3. The case m = 2 is known to be impossible [Aub82].\nThe even case m > 2 remains open.
\n [Knu26] D. E. Knuth, \"Claude's Cycles\" (2026).
\n\n [Aub82] J. Aubert, B. Schneider, \"Graphes orientés indécomposables en circuits hamiltoniens\",\nJ. Combin. Theory Ser. B 32 (1982), 347–349.
\n\n A polyhedron is Rupert if one can cut a hole in it and pass another\ncopy of the same polyhedron through that hole.
\n\n More formally: a convex body in ℝ³ is a compact, convex set with\nnonempty interior. A convex body X is said to be Rupert if there are\ntwo affine transforms T₁, T₂ ∈ SE(3) such that π(T₁(X)) ⊆\nint(π(T₂(X))), where π : ℝ³ → ℝ² is the evident projection, and int\ndenotes topological interior.
\n\n Not all convex bodies are Rupert. For example,
\n\n the unit ball is not Rupert
\n\n the circular cylinder of unit diameter and height\nclosed on each end by disks is not Rupert
\n\n However, many convex polyhedra are Rupert. All Platonic solids, and\nmost Archimedean and Catalan solids are known to be Rupert.
\n\n Question: are all convex polyhedra with nonempty interior Rupert?
\n\n
\nPlatonic Passages,\nR. P. Jerrard, J. E. Wetzel, and L. Yuan., Math. Mag., 90(2):87–98,\n2017. conjectures (\"with a certain hesitancy\") that perhaps all\nconvex polyhedra are Rupert.
\n\n However, An Algorithmic Approach to Rupert's Problem\ndescribes experimental evidence to suggest that three Archimedean\nsolids may not be Rupert.
\n\nOptimizing for the Rupert property\nis the source of some of the Catalan solid results, and has more\nresults for Johnson polyhedra as well.
\n\nThis video by David Renshaw visualizes\nknown results for Platonic, Archimedean, and Catalan solids.
\n\n This problem's name comes from the fact that it is a generalization\nof Prince Rupert's Cube.
\n\nA convex polyhedron without Rupert's property,\nJakob Steininger and Sergey Yurkevich, 2025. Constructs a convex polyhedron and\na proof that it is not Rupert, resolving the open question.
\n\n The Bézier-type Bernstein operators $B_{n,\\alpha}$ for $\\alpha > 0$ are defined for\n$f : [0,1] \\to \\mathbb{R}$ by\n[\n(B_{n,\\alpha} f)(x)\n= \\sum_{k=0}^n f!\\left(\\frac{k}{n}\\right)\n\\left( J_{n,k}(x)^{\\alpha} - J_{n,k+1}(x)^{\\alpha} \\right),\n]\nwhere\n[\nJ_{n,k}(x) = \\sum_{j=k}^n p_{n,j}(x),\n\\qquad\np_{n,j}(x) = \\binom{n}{j} x^j(1-x)^{n-j},\n]\nand $J_{n,n+1}(x) = 0$.
\n\n In the classical case $\\alpha = 1$, these operators reduce to the usual Bernstein operators.\nFor $f$ which are $C^2$ on $[0,1]$, one has the classical Voronovskaja\nasymptotic formula\n[\n\\lim_{n \\to \\infty} n\\bigl( B_{n,1} f(x) - f(x) \\bigr)\n= \\tfrac{1}{2} x(1-x) f''(x).\n]
\n\n For $\\alpha = 1$, the asymptotics are completely understood.
\n\n Numerical experiments indicate that for $\\alpha \\neq 1$ the quantity\n[\n\\sqrt{n},\\bigl( B_{n,\\alpha} f(x) - f(x) \\bigr)\n]\nmay converge to a non-zero limit.
\n\n Determine the asymptotic behaviour of the Bézier-type Bernstein operators for $\\alpha > 0$,\n$\\alpha \\neq 1$:\n\\textbf{Existence of the limit:}\nProve (or disprove) the existence of the limit\n[\n\\lim_{n \\to \\infty}\n\\sqrt{n},\\bigl( B_{n,\\alpha} f(x) - f(x) \\bigr),\n]\nat least for sufficiently smooth functions $f$.\n\\textbf{Explicit form of the limit:}\nIf the limit exists, determine an explicit expression for it in terms of $f$, $x$, and $\\alpha$.
\n\n
\nVoronovskaja-type Formula for the Bézier Variant of the Bernstein Operators,\nby
\n Problems 4.1, 4.2, and 4.3 from arxiv/2506.23631.
\n\n
\n See also FormalConjectures.Wikipedia.Fuglede for Fuglede's spectral set conjecture, which\nmotivates the study of weak tilings.
\n
\nAbout a New Kind of Ramanujan-Type Series by
\n [G2003] Guillera, Jesús. \"About a new kind of Ramanujan-type series.\" Experimental Mathematics 12.4 (2003): 507-510.
\n\n [A2025] Au, Kam Cheong. \"Wilf-Zeilberger seeds and non-trivial hypergeometric identities.\" Journal of Symbolic Computation 130 (2025): 102421. arXiv:2312.14051
\n\n
\n The Latin Tableau Conjecture states that the graph associated\nto any (finite) Young diagram (i.e., whose vertices are the\ncells of the diagram, with edges between cells in the same row\nor column) is CDS-colorable, meaning that there exists a proper\ncoloring of the vertices of the graph such that for all k > 0, the\nnumber of vertices with color < k equals the maximum size of\nthe union of k independent sets of the graph.
\n\n
\n
\nB. Reed, ω Δ and χ, J. Graph Theory 27 (1998) 177-212.
\n\n\nmathoverflow/37923 asked by user Andrew D. King
\nbs(f) ≤ s(f)^2)\n This file formalizes the
\n For every Boolean function f : {0,1}^n → {0,1},\nbs(f) ≤ s(f)^2,\nwhere bs(f) denotes block sensitivity and s(f) denotes sensitivity.
\n Huang's theorem proves a bs(f) ≤ s(f)^4, thereby\nresolving the most widely known form of the sensitivity conjecture.
\n We now ask whether a stronger upper bound holds. Interestingly, the original\npaper of Nisan and Szegedy, where the sensitivity conjecture first appeared,\nalready speculated that a bs(f) ≥ (2/3)⋅s(f)^2.
\n
\nInduced Subgraphs of Hypercubes and a Proof of the Sensitivity Conjecture\nby Hao Huang (see Section 3, Concluding Remarks)
\n\nVariations on the Sensitivity Conjecture\nby Pooya Hatami, Raghav Kulkarni, and Denis Pankratov (see Question 3.1)
\n\nOn the Degree of Boolean Functions as Real Polynomials\nby Noam Nisan, and Mario Szegedy (see Section 4, Open Problems)
\n\n
\nFusible numbers and Peano Arithmetic,\nby Jeff Erickson, Gabriel Nivasch, and Junyan Xu.
\n\nFusible numbers and Peano Arithmetic,\nLogical Methods in Computer Science, Volume 18, Issue 3 (July 28, 2022).
\n\n This file formalizes the notion of a weakly first countable topological space and some conjectures\naround those.
\n\n
\n [Ar2013] Arhangeliski, Alexandr. \"Selected old open problems in general topology.\"\nBuletinul Academiei de Ştiinţe a Republicii Moldova. Matematica 73.2-3 (2013): 37-46.\nhttps://www.math.md/files/basm/y2013-n2-3/y2013-n2-3-(pp37-46).pdf.pdf
\n\n [Ya1976] Yakovlev, N. N. \"On the theory of o-metrizable spaces.\"\nDoklady Akademii Nauk. Vol. 229. No. 6. Russian Academy of Sciences, 1976.\nhttps://www.mathnet.ru/links/016f74007f9f96fa3aadae05cbd98457/dan40570.pdf (in Russian)
\n\n
\n The conjecture asks for a Lindelöf space where all singletons are G_δ sets\nand which has cardinality > 𝔠.
\n\n This is Problem 1 in https://www.math.md/files/basm/y2013-n2-3/y2013-n2-3-(pp37-46).pdf.pdf
\n\n
\nSelected Old Open Problems in General Topology\nby A. V. Arhangel’skii
\n\n
\n This file formalizes the notion of a weakly first countable topological space and some conjectures\naround those.
\n\n
\n [Ar2013] Arhangeliski, Alexandr. \"Selected old open problems in general topology.\"\nBuletinul Academiei de Ştiinţe a Republicii Moldova. Matematica 73.2-3 (2013): 37-46.\nhttps://www.math.md/files/basm/y2013-n2-3/y2013-n2-3-(pp37-46).pdf.pdf
\n\n This file formalizes some conjectures and theorems around latin squares.
\n\n
\n [Wa2011] Wanless, Ian. \"Transversals in Latin Squares: A Survey.\"\nSurveys in Combinatorics 2011, R. Chapman, Ed. Cambridge University Press, 2011, pp. 403–437.\nhttps://users.monash.edu.au/~iwanless/papers/transurveyBCC.pdf
\n\n https://en.wikipedia.org/wiki/Problems_in_Latin_squares
\n\n
\n [Za94] Zagier, Don. \"Values of zeta functions and their applications.\"\nFirst European Congress of Mathematics Paris, July 6–10, 1992: Vol. II: Invited Lectures (Part 2). Basel: Birkhäuser Basel, 1994.
\n\n [Co18] Combariza, Germán AG. \"A few conjectures about the multiple zeta values.\"\nACM Communications in Computer Algebra 52.1 (2018): 11-20.
\n\n [Te02] T. Terasoma. Mixed Tate motives and multiple zeta values. Invent. Math., 149(2):339–369, 2002.
\n\n [DG05] P. Deligne and A. Goncharov. Groupes fondamentaux motiviques de Tate mixte. Ann. Sci.\nEcole Norm. Sup. (4), 38(1):1–56, 2005.
\n\n\n
\n
\n The Casas-Alvero conjecture states that if a univariate polynomial P of degree d over a field\nof characteristic zero shares a non-trivial factor with its Hasse derivatives up to order d-1,\nthen P must be of the form (X - α)ᵈ for some α in the field.
\n The conjecture has been proven for:
\n\n Degrees d ≤ 8
\n Degrees of the form p^k where p is prime
\n Degrees of the form 2p^k where p is prime
\n The conjecture is false in positive characteristic p for polynomials of degree p+1.
\n The conjecture is now claimed to be proven in this paper:
\n","/FormalConjectures/Paper/HartshorneConjecture/":"\n
\n [Har1974] R. Hartshorne, Varieties of small codimension in projective space.
\n\n [MO2010] Evidences on Hartshorne's conjecture? References?
\n\n The game Catch-Up (Isaksen–Ismail–Brams–Nealen, 2015) is a two-player, perfect-information game\nplayed on a finite nonempty set S of positive integers. Each time a player removes a number from\nS, that number is added to the player’s score.
\nRules.
\n\n The scores start at 0. Player p1 starts by removing exactly one number from S.
\n After the first move, players alternate turns. On a turn, the current player removes one or more\nnumbers from S, one at a time, and must keep removing numbers until their score becomes\nat least the opponent’s score; before the final pick they must remain strictly behind.
\n If the current player cannot catch up (in particular, even taking all remaining numbers would still\nleave them behind), the game ends immediately: the current player receives all remaining numbers.
\n\n When S is empty, the player with higher score wins; equal scores give a draw.
\n In this file we define:
\n\nPlayer and Outcome,
\n the recursive evaluator value (optimal play),
\n the conjecture value_of_even_mul_succ_self_div_two.
\n For S = {1,2,3,4} one play is: p1 takes 2, p2 takes 1 then 4, and p1 takes 3,\nending with scores (5,5).
\n A. Isaksen, M. Ismail, S. J. Brams, A. Nealen,\n
\n Title: Degree sequences in triangle-free graphs\nAuthors: P. Erdős, S. Fajtlowicz and W. Staton,\nPublished in Discrete Mathematics 92 (1991) 85–88.
","/FormalConjectures/OEIS/«287616»/":"\n Any nonnegative integer can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ with $x, y, z$\nnonnegative integers.
\n\n Zhi-Wei Sun has offered a USD 135 prize for the first proof of this conjecture.
\n\n
\nA287616
\n\n Zhi-Wei Sun, \"Universal sums of three quadratic polynomials\", arXiv:1502.03056 [math.NT]
\n\n Informal Statement:\nFor an integer $k ≥ 2$, the following are equivalent:
\n\n The greatest common divisor of the binomial coefficients\n$\\binom{2k}{k}, \\binom{3k}{k}, \\dots, \\binom{(k+1)k}{k} = 1$.
\n\n Writing prime factorization of k as\n$k = \\prod p_i^{e_i}$, and let\n$P = \\max_i p_i^{e_i}$,\none has $k / P > P$.
\n\n This conjecture asserts that the sequence defined by 1. is obtained by\ntaking 1 off each number in the sequence defined by 2.
\n\n
\n Any nonnegative integer can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative\nintegers such that $x^4 + 1680 y^3 z$ is a square.
\n\n Zhi-Wei Sun has offered a prize of 1,680 RMB for the first proof.
\n\n
\nA280831
\n\n Z.-W. Sun, \"Refining Lagrange's four-square theorem,\"
\n Z.-W. Sun, \"Refining Lagrange's four-square theorem,\" arXiv:1604.06723 [math.NT], 2016.
\n\n Let $a_n$ be the number of distinct subwords (contiguous factors) of length $n$ in the infinite\nword generated by the morphism $\\sigma: a \\mapsto aab, b \\mapsto b$, starting from $a$.\nAs shown in the references, $a_n$ is given by the formula\n$$ a_n = \\sum_{i=0}^{n} \\min(2^i,n-i+1). $$
\n\n The conjectured generating function is:\n$$\\sum_{n \\geq 0} a_n x^n = \\frac{1}{1-x} + \\frac{x}{(1-x)^2}\\left(\\frac{1}{1-x} -\n\\sum_{k \\geq 1} x^{2^k + k - 1}\\right).$$
\n\n
\nA6697
\n\n J.-P. Allouche and J. Shallit, \"On the subword complexity of the fixed point of a → aab, b → b,\nand generalizations,\" arXiv:1605.02361 [math.CO], 2016.
\n\n N. J. A. Sloane and Simon Plouffe,
\n Any integer $n > 0$ can be written as $\\binom{w+2}{2} + \\binom{x+3}{4} + \\binom{y+5}{6} + \\binom{z+7}{8}$\nwith $w, x, y, z$ nonnegative integers.
\n\n Zhi-Wei Sun has offered a $2,468 prize for the first proof (or $2,468 RMB for a counterexample).
\n\n The conjecture has been verified for all $n$ up to $1.2 \\times 10^{12}$ by Yaakov Baruch (March 2019).
\n\n
\nA306477
\n\nmathoverflow/323541: Z.-W. Sun, \"Positive integers written as C(w,2) + C(x,4) + C(y,6) + C(z,8) with w,x,y,z in {2,3,...},\", Feb. 19, 2019.
\n\n Smallest number $k$ such that $kn + 1$ is prime.
\n\n
\n Every prime $p$ has a primitive root $0 < g < p$ of the form $k^2 + 1$, where $k$ is an integer.
\n\n Zhi-Wei Sun has offered a prize of RMB 2,000 for the first proof.
\n\n
\nA239957
\n\n Z.-W. Sun, \"New observations on primitive roots modulo primes,\" arXiv:1405.0290 [math.NT], 2014.
\n\n Numbers n such that $n^2 + \\pi(n)$ is prime.
\n\n
\n A063880 lists numbers $n$ such that $\\sigma(n) = 2 \\cdot \\text{usigma}(n)$, where $\\sigma(n)$ is the\nsum of all divisors and $\\text{usigma}(n)$ is the sum of unitary divisors.
\n\n Equivalently, these are numbers whose unitary and non-unitary divisors have equal sum.
\n\n The conjectures state that all members satisfy $n \\equiv 108 \\pmod{216}$, and that all\nprimitive terms (those whose proper divisors aren't in the sequence) are powerful numbers,\nwith $108$ being the only primitive term.
\n\n
\n Name: \"No powers as partition numbers\"
\n\n There are no partition numbers $p(k)$ of the form $x^m$, with $x,m$ integers $>1$.
\n\n
\n Define $\\varsigma(n)$ the smallest prime factor of $n$ (Nat.minFac). Let $a_n$ be the least\nnumber such that the count of numbers $k \\le a_n$ with $k > \\varsigma(k)^n$ exceeds the count\nof numbers with $k \\le \\varsigma(k)^n$.
\n The conjecture states that $a_n = 3^n + 3 \\cdot 2^n + 6$ for $n \\ge 1$.
\n\n
\n A067720 lists numbers $k$ such that $\\varphi(k^2 + 1) = k \\cdot \\varphi(k + 1)$,\nwhere $\\varphi$ is Euler's totient function.
\n\n The sequence exhibits a strong connection to primes: for almost all terms $k$,\n$k + 1$ is prime. The conjecture states that $k = 8$ is the only exception.
\n\n
\n Any integer $n \\geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative\nintegers and $z \\leq w$, such that both $x$ and $x + 24y$ are squares.
\n\n Zhi-Wei Sun has offered a $2,400 prize for the first proof.
\n\n
\nA281976
\n\n Z.-W. Sun, \"Refining Lagrange's four-square theorem,\"
\n Z.-W. Sun, \"Restricted sums of four squares,\"
\n There are infinite primes of the form $2^n + 2^i + 1$, with $0 < i < n$.\nSee Wagstaff (2001) where this conjecture is posed.
\n\n
\n Samuel S. Wagstaff, Jr., Prime Numbers with a fixed number of one bits or zero bits in their binary\nrepresentation, Exp. Math. vol. 10, issue 2 (2001) 267.
\n\nA81091
\n\n A56777 lists composite numbers $n$ satisfying both $\\varphi(n+12) = \\varphi(n) + 12$ and\n$\\sigma(n+12) = \\sigma(n) + 12$.
\n\n The conjectures state identities connecting A56777 and prime quadruples (A7530), as\nwell as congruences satisfied by the members of A56777.
\n\n
\n Number of ways to write $n = x+y$, for $x,y > 0$ such that $2^x + y$ is prime.
\n\n Zhi-Wei Sun has offered a $1000 prize for the first proof.
\n\n
\nA231201
\n\n Zhi-Wei Sun, \"Table of n, a(n) for n = 1..10000\",\n\"Write n = k + m with 2^k + m prime\", a message to Number Theory List, Nov. 16, 2013,\n\"On a^n+ bn modulo m\", arXiv:1312.1166 [math.NT], 2013-2014,\n\"Problems on combinatorial properties of primes\", arXiv:1402.6641 [math.NT], 2014-2015.
\n\n Any integer $n > 1$ can be written as $a^2 + b^2 + 3^c + 5^d$ where $a, b, c, d$ are\nnonnegative integers.
\n\n Zhi-Wei Sun has offered a $3,500 prize for the first proof.
\n\n
\nA303656
\n\n Z.-W. Sun, \"Restricted sums of four squares,\" arXiv preprint:\nhttps://arxiv.org/abs/1701.05868v10
\n\n Z.-W. Sun, \"Refining Lagrange's four-square theorem,\" Journal of Number Theory:\nhttp://maths.nju.edu.cn/~zwsun/RefineFourSquareTh.pdf
\n\n Z.-W. Sun, \"Restricted sums of three or four squares\":\nhttp://maths.nju.edu.cn/~zwsun/Square-sum.pdf
\n\n Zhi-Wei Sun's 1-3-5 conjecture and variations:\nhttps://www.aimspress.com/aimspress-data/era/2020/2/PDF/1935-9179_2020_2_589.pdf
\n\n Numerator of $sum_{k = 1}^n \\frac{1}{k^3} * \\binom{n}{k}^2 * \\binom{n+k}{k}^2 for $n \\ge 1$\nwith $a(0) = 0$.
\n\n
\n Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and\n$x^2 + ny^2$ are prime.
\n\n Zhi-Wei Sun has offered a $200 prize for the first proof.
\n\n
\nA232174
\n\n Z.-W. Sun, \"Conjectures on representations involving primes,\" in: M. Nathanson (ed.),\nCombinatorial and Additive Number Theory II: CANT, Springer Proc. in Math. & Stat.,\nVol. 220, Springer, 2017, pp. 279-310. https://arxiv.org/abs/1211.1588
\n\n D.A. Cox, \"Primes of the Form x² + ny²,\" John Wiley & Sons, 1989.
\n\n $a(n)$ is the minimum integer $k$ such that the smallest prime factor of the\n$n$-th Fermat number exceeds $2^(2^n - k)$.
\n\n
\nA358684
\n\nSA22 Lorenzo Sauras-Altuzarra,
\n Any integer $n > 1$ can be written as $(2^a \\cdot 3^b)^2 + (2^c \\cdot 5^d)^2 + x^2 + y^2$\nwhere $a, b, c, d, x, y$ are nonnegative integers.
\n\n Zhi-Wei Sun has offered a $2,500 prize for the first proof.
\n\n
\nA308734
\n\n Z.-W. Sun, \"Refining Lagrange's four-square theorem,\"
\n Z.-W. Sun, \"Restricted sums of four squares,\"
\n Z.-W. Sun, \"Various Refinements of Lagrange's Four-Square Theorem,\" Westlake Number Theory\nSymposium, Nanjing University, China, 2020.
\n\n S. Banerjee, \"On a conjecture of Sun about sums of restricted squares,\"
\n A random subset of 100 open research problems, drawn uniformly at random\nfrom all problems with the category research open tag.
\n A random subset of 100 non-open problems, drawn uniformly at random\nfrom all problems without the category research open tag\n(solved, test, API, etc.).
\n The type of formal proof that exists for a problem.
"},"ProblemAttributes.FormalProofKind.formalConjecturesProof":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___FormalProofKind___formalConjecturesProof","anchor":"ProblemAttributes___FormalProofKind___formalConjecturesProof","docHtml":"\n The problem exactly as stated in formal-conjectures has a formal proof.\nThe link points to a commit that fills the sorry relative to the current\ncommit (i.e., the commit where this category is added, or the commit with the\nlatest fix for this statement).
\n The problem is solved in Lean 4 (e.g. in Mathlib or some other\nrepository), perhaps as an equivalent statement.
"},"ProblemAttributes.FormalProofKind.otherSystem":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___FormalProofKind___otherSystem","anchor":"ProblemAttributes___FormalProofKind___otherSystem","docHtml":"\n The problem is formally solved in a different system (Roqc, Isabelle, Lean 3, HOL, etc.).
"},"ProblemAttributes.ProblemStatus":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___ProblemStatus","anchor":"ProblemAttributes___ProblemStatus"},"ProblemAttributes.ProblemStatus.open":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___ProblemStatus___open","anchor":"ProblemAttributes___ProblemStatus___open","docHtml":"\n Indicates that a mathematical problem is still open.
"},"ProblemAttributes.ProblemStatus.solved":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___ProblemStatus___solved","anchor":"ProblemAttributes___ProblemStatus___solved","docHtml":"\n Indicates that a mathematical problem is already solved,\ni.e., there is a published (informal) proof that is widely accepted by experts.
"},"ProblemAttributes.formalProofKind":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___formalProofKind","anchor":"ProblemAttributes___formalProofKind"},"ProblemAttributes.formalProofKind.toName":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___formalProofKind___toName","anchor":"ProblemAttributes___formalProofKind___toName"},"ProblemAttributes.problemStatus":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___problemStatus","anchor":"ProblemAttributes___problemStatus"},"ProblemAttributes.problemStatus.toName":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___problemStatus___toName","anchor":"ProblemAttributes___problemStatus___toName","docHtml":"\n Convert from a syntax node to a name.
"},"ProblemAttributes.Category":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___Category","anchor":"ProblemAttributes___Category","docHtml":"\n A type to capture the various types of statements that appear in our Lean files.
"},"ProblemAttributes.Category.textbook":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___Category___textbook","anchor":"ProblemAttributes___Category___textbook","docHtml":"\n A textbook level math problem (high school, undergraduate, or graduate).
"},"ProblemAttributes.Category.research":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___Category___research","anchor":"ProblemAttributes___Category___research","docHtml":"\n A reseach level math problem. This can be open, or already solved
"},"ProblemAttributes.Category.test":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___Category___test","anchor":"ProblemAttributes___Category___test","docHtml":"\n A test statement that serves as a sanity check (e.g. for a new definition)
"},"ProblemAttributes.Category.API":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___Category___API","anchor":"ProblemAttributes___Category___API","docHtml":"\n An \"API\" statement, i.e. a statement that constructs basic theory around a new definition
"},"ProblemAttributes.CategorySyntax":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___CategorySyntax","anchor":"ProblemAttributes___CategorySyntax"},"ProblemAttributes.CategoryTag":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___CategoryTag","anchor":"ProblemAttributes___CategoryTag"},"ProblemAttributes.CategoryTag.declName":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___CategoryTag___declName","anchor":"ProblemAttributes___CategoryTag___declName","docHtml":"\n The name of the declaration with the given tag.
"},"ProblemAttributes.CategoryTag.category":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___CategoryTag___category","anchor":"ProblemAttributes___CategoryTag___category","docHtml":"\n The status of the problem.
"},"ProblemAttributes.CategoryTag.informal":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___CategoryTag___informal","anchor":"ProblemAttributes___CategoryTag___informal","docHtml":"\n The (optional) comment that comes with the given declaration.
"},"ProblemAttributes.categoryExt":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___categoryExt","anchor":"ProblemAttributes___categoryExt","docHtml":"\n Defines the categoryExt extension for adding a HashSet of Tags\nto the environment.
\n A tag recording the existence and location of a formal proof for a declaration.
"},"ProblemAttributes.FormalProofTag.declName":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___FormalProofTag___declName","anchor":"ProblemAttributes___FormalProofTag___declName","docHtml":"\n The name of the declaration with the given tag.
"},"ProblemAttributes.FormalProofTag.proofKind":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___FormalProofTag___proofKind","anchor":"ProblemAttributes___FormalProofTag___proofKind","docHtml":"\n The kind of formal proof.
"},"ProblemAttributes.FormalProofTag.proofLink":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___FormalProofTag___proofLink","anchor":"ProblemAttributes___FormalProofTag___proofLink","docHtml":"\n A link to the formal proof.
"},"ProblemAttributes.formalProofExt":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___formalProofExt","anchor":"ProblemAttributes___formalProofExt","docHtml":"\n Defines the formalProofExt extension for recording formal proof annotations.
\n The name of the declaration with the given tag.
"},"ProblemAttributes.SubjectTag.subjects":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___SubjectTag___subjects","anchor":"ProblemAttributes___SubjectTag___subjects","docHtml":"\n The subject(s) of the problem.
"},"ProblemAttributes.SubjectTag.informal":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___SubjectTag___informal","anchor":"ProblemAttributes___SubjectTag___informal","docHtml":"\n The (optional) comment that comes with the given declaration.
"},"ProblemAttributes.subjectExt":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___subjectExt","anchor":"ProblemAttributes___subjectExt","docHtml":"\n Defines the tagExt extension for adding a HashSet of Tags\nto the environment.
\n Convert from a syntax node to a term of type Category and annotate the syntax\nwith the corresponding name's docstring.
\n Converts a syntax node to an array of AMS subjects.
\n This also annotates the every natural number litteral encountered, with the\ndescription of the corresponding AMS subject (i.e. hovering over the number\nin VS Code will show the subject.)
"},"ProblemAttributes.problemSubject":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___problemSubject","anchor":"ProblemAttributes___problemSubject"},"ProblemAttributes.splitByFun":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___splitByFun","anchor":"ProblemAttributes___splitByFun","docHtml":"\n Split an array into preimages of a function.
\n\nsplitByFun f arr is the hashmap such that the value for\nkey b : β is the array of a : α in arr that get mapped\nto b by f
\n Split an array into preimages of a function.
\n\nsplitByFun f arr is the hashmap such that the value for\nkey b : β is the array of a : α in arr that get mapped\nto b by f
\n Split an array into preimages of a function.
\n\nsplitByFun f arr is the hashmap such that the value for\nkey b : β is the array of a : α in arr that get mapped\nto b by f
\n Get the formal proof tag for a given declaration, if any.
"},"ProblemAttributes.verifyCategoryCounts":{"url":"/FormalConjectures/Util/Attributes/Basic/#ProblemAttributes___verifyCategoryCounts","anchor":"ProblemAttributes___verifyCategoryCounts","docHtml":"\n Verify that the list of problems contains the expected number of problems\nfor each category. Throws an error if counts do not match.
"},"AMS":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS","anchor":"AMS","docHtml":"\n This file defines some tools used by the ProblemSubject attribute in order classify\nproblems by their corresponding AMS Subject.
\n The AMSDescription has one term for each number n ∈ {1, ..., 96} that has a corresponding\nAMS subject, namely AMSDescription.«n». Note that not all values of n in this interval\nare assigned a subject.
\n To extract the value corresponding to n, one can use numToAMSDescriptions n. This is useful\nfor getting the doctring that corresponds to the subject n when parsing the attribute.
\n Finally, to access the list of subjects and their corresponding number when editing Lean files,\nwe implement a #AMS command that prints this list.
\n General and overarching topics
"},"AMS.«1»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_1_FLQQ_","anchor":"AMS____FLQQ_1_FLQQ_","docHtml":"\n History and biography
"},"AMS.«3»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_3_FLQQ_","anchor":"AMS____FLQQ_3_FLQQ_","docHtml":"\n Mathematical logic and foundations
"},"AMS.«5»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_5_FLQQ_","anchor":"AMS____FLQQ_5_FLQQ_","docHtml":"\n Combinatorics
"},"AMS.«6»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_6_FLQQ_","anchor":"AMS____FLQQ_6_FLQQ_","docHtml":"\n Order, lattices, ordered algebraic structures
"},"AMS.«8»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_8_FLQQ_","anchor":"AMS____FLQQ_8_FLQQ_","docHtml":"\n General algebraic systems
"},"AMS.«11»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_11_FLQQ_","anchor":"AMS____FLQQ_11_FLQQ_","docHtml":"\n Number theory
"},"AMS.«12»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_12_FLQQ_","anchor":"AMS____FLQQ_12_FLQQ_","docHtml":"\n Field theory and polynomials
"},"AMS.«13»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_13_FLQQ_","anchor":"AMS____FLQQ_13_FLQQ_","docHtml":"\n Commutative algebra
"},"AMS.«14»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_14_FLQQ_","anchor":"AMS____FLQQ_14_FLQQ_","docHtml":"\n Algebraic geometry
"},"AMS.«15»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_15_FLQQ_","anchor":"AMS____FLQQ_15_FLQQ_","docHtml":"\n Linear and multilinear algebra; matrix theory
"},"AMS.«16»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_16_FLQQ_","anchor":"AMS____FLQQ_16_FLQQ_","docHtml":"\n Associative rings and algebras
"},"AMS.«17»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_17_FLQQ_","anchor":"AMS____FLQQ_17_FLQQ_","docHtml":"\n Nonassociative rings and algebras
"},"AMS.«18»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_18_FLQQ_","anchor":"AMS____FLQQ_18_FLQQ_","docHtml":"\n Category theory; homological algebra
"},"AMS.«19»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_19_FLQQ_","anchor":"AMS____FLQQ_19_FLQQ_","docHtml":"\n K-theory
"},"AMS.«20»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_20_FLQQ_","anchor":"AMS____FLQQ_20_FLQQ_","docHtml":"\n Group theory and generalizations
"},"AMS.«22»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_22_FLQQ_","anchor":"AMS____FLQQ_22_FLQQ_","docHtml":"\n Topological groups, Lie groups
"},"AMS.«26»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_26_FLQQ_","anchor":"AMS____FLQQ_26_FLQQ_","docHtml":"\n Real functions
"},"AMS.«28»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_28_FLQQ_","anchor":"AMS____FLQQ_28_FLQQ_","docHtml":"\n Measure and integration
"},"AMS.«30»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_30_FLQQ_","anchor":"AMS____FLQQ_30_FLQQ_","docHtml":"\n Functions of a complex variable
"},"AMS.«31»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_31_FLQQ_","anchor":"AMS____FLQQ_31_FLQQ_","docHtml":"\n Potential theory
"},"AMS.«32»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_32_FLQQ_","anchor":"AMS____FLQQ_32_FLQQ_","docHtml":"\n Several complex variables and analytic spaces
"},"AMS.«33»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_33_FLQQ_","anchor":"AMS____FLQQ_33_FLQQ_","docHtml":"\n Special functions
"},"AMS.«34»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_34_FLQQ_","anchor":"AMS____FLQQ_34_FLQQ_","docHtml":"\n Ordinary differential equations
"},"AMS.«35»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_35_FLQQ_","anchor":"AMS____FLQQ_35_FLQQ_","docHtml":"\n Partial differential equations
"},"AMS.«37»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_37_FLQQ_","anchor":"AMS____FLQQ_37_FLQQ_","docHtml":"\n Dynamical systems and ergodic theory
"},"AMS.«39»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_39_FLQQ_","anchor":"AMS____FLQQ_39_FLQQ_","docHtml":"\n Difference and functional equations
"},"AMS.«40»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_40_FLQQ_","anchor":"AMS____FLQQ_40_FLQQ_","docHtml":"\n Sequences, series, summability
"},"AMS.«41»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_41_FLQQ_","anchor":"AMS____FLQQ_41_FLQQ_","docHtml":"\n Approximations and expansions
"},"AMS.«42»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_42_FLQQ_","anchor":"AMS____FLQQ_42_FLQQ_","docHtml":"\n Harmonic analysis on Euclidean spaces
"},"AMS.«43»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_43_FLQQ_","anchor":"AMS____FLQQ_43_FLQQ_","docHtml":"\n Abstract harmonic analysis
"},"AMS.«44»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_44_FLQQ_","anchor":"AMS____FLQQ_44_FLQQ_","docHtml":"\n Integral transforms, operational calculus
"},"AMS.«45»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_45_FLQQ_","anchor":"AMS____FLQQ_45_FLQQ_","docHtml":"\n Integral equations
"},"AMS.«46»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_46_FLQQ_","anchor":"AMS____FLQQ_46_FLQQ_","docHtml":"\n Functional analysis
"},"AMS.«47»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_47_FLQQ_","anchor":"AMS____FLQQ_47_FLQQ_","docHtml":"\n Operator theory
"},"AMS.«49»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_49_FLQQ_","anchor":"AMS____FLQQ_49_FLQQ_","docHtml":"\n Calculus of variations and optimal control; optimization
"},"AMS.«51»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_51_FLQQ_","anchor":"AMS____FLQQ_51_FLQQ_","docHtml":"\n Geometry
"},"AMS.«52»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_52_FLQQ_","anchor":"AMS____FLQQ_52_FLQQ_","docHtml":"\n Convex and discrete geometry
"},"AMS.«53»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_53_FLQQ_","anchor":"AMS____FLQQ_53_FLQQ_","docHtml":"\n Differential geometry
"},"AMS.«54»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_54_FLQQ_","anchor":"AMS____FLQQ_54_FLQQ_","docHtml":"\n General topology
"},"AMS.«55»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_55_FLQQ_","anchor":"AMS____FLQQ_55_FLQQ_","docHtml":"\n Algebraic topology
"},"AMS.«57»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_57_FLQQ_","anchor":"AMS____FLQQ_57_FLQQ_","docHtml":"\n Manifolds and cell complexes
"},"AMS.«58»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_58_FLQQ_","anchor":"AMS____FLQQ_58_FLQQ_","docHtml":"\n Global analysis, analysis on manifolds
"},"AMS.«60»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_60_FLQQ_","anchor":"AMS____FLQQ_60_FLQQ_","docHtml":"\n Probability theory and stochastic processes
"},"AMS.«62»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_62_FLQQ_","anchor":"AMS____FLQQ_62_FLQQ_","docHtml":"\n Statistics
"},"AMS.«65»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_65_FLQQ_","anchor":"AMS____FLQQ_65_FLQQ_","docHtml":"\n Numerical analysis
"},"AMS.«68»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_68_FLQQ_","anchor":"AMS____FLQQ_68_FLQQ_","docHtml":"\n Computer science
"},"AMS.«70»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_70_FLQQ_","anchor":"AMS____FLQQ_70_FLQQ_","docHtml":"\n Mechanics of particles and systems
"},"AMS.«74»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_74_FLQQ_","anchor":"AMS____FLQQ_74_FLQQ_","docHtml":"\n Mechanics of deformable solids
"},"AMS.«76»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_76_FLQQ_","anchor":"AMS____FLQQ_76_FLQQ_","docHtml":"\n Fluid mechanics
"},"AMS.«78»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_78_FLQQ_","anchor":"AMS____FLQQ_78_FLQQ_","docHtml":"\n Optics, electromagnetic theory
"},"AMS.«80»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_80_FLQQ_","anchor":"AMS____FLQQ_80_FLQQ_","docHtml":"\n Classical thermodynamics, heat transfer
"},"AMS.«81»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_81_FLQQ_","anchor":"AMS____FLQQ_81_FLQQ_","docHtml":"\n Quantum theory
"},"AMS.«82»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_82_FLQQ_","anchor":"AMS____FLQQ_82_FLQQ_","docHtml":"\n Statistical mechanics, structure of matter
"},"AMS.«83»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_83_FLQQ_","anchor":"AMS____FLQQ_83_FLQQ_","docHtml":"\n Relativity and gravitational theory
"},"AMS.«85»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_85_FLQQ_","anchor":"AMS____FLQQ_85_FLQQ_","docHtml":"\n Astronomy and astrophysics
"},"AMS.«86»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_86_FLQQ_","anchor":"AMS____FLQQ_86_FLQQ_","docHtml":"\n Geophysics
"},"AMS.«90»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_90_FLQQ_","anchor":"AMS____FLQQ_90_FLQQ_","docHtml":"\n Operations research, mathematical programming
"},"AMS.«91»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_91_FLQQ_","anchor":"AMS____FLQQ_91_FLQQ_","docHtml":"\n Game theory, economics, social and behavioral sciences
"},"AMS.«92»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_92_FLQQ_","anchor":"AMS____FLQQ_92_FLQQ_","docHtml":"\n Biology and other natural sciences
"},"AMS.«93»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_93_FLQQ_","anchor":"AMS____FLQQ_93_FLQQ_","docHtml":"\n Systems theory; control
"},"AMS.«94»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_94_FLQQ_","anchor":"AMS____FLQQ_94_FLQQ_","docHtml":"\n Information and communication, circuits
"},"AMS.«97»":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS____FLQQ_97_FLQQ_","anchor":"AMS____FLQQ_97_FLQQ_","docHtml":"\n Mathematics education
"},"numToAMSName":{"url":"/FormalConjectures/Util/Attributes/AMS/#numToAMSName","anchor":"numToAMSName"},"AMS.getDesc":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS___getDesc","anchor":"AMS___getDesc"},"AMS.toNat?":{"url":"/FormalConjectures/Util/Attributes/AMS/#AMS___toNat___","anchor":"AMS___toNat___"},"numToAMSSubjects":{"url":"/FormalConjectures/Util/Attributes/AMS/#numToAMSSubjects","anchor":"numToAMSSubjects"},"linter.style.ams_attribute":{"url":"/FormalConjectures/Util/Linters/AMSLinter/#linter___style___ams_attribute","anchor":"linter___style___ams_attribute","docHtml":"\n The AMSLinter is a linter to aid with formatting contributions to\nthe Formal Conjectures repository by ensuring that results in a file have\nthe appropriate subject tags.
\n Checks if a command has the AMS attribute.
\n The problem category linter checks that every theorem/lemma/example\nhas been given an AMS attribute.
\n This file contains test cases for the CategoryDocstringLinter, verifying that\nresearch-open, research-solved, and textbook declarations without docstrings are flagged,\nwhile declarations with docstrings or other categories are accepted.
\n A documented open problem statement should not be flagged.
"},"CategoryDocstringLinter.not_flagged_test_without_docstring":{"url":"/FormalConjectures/Util/Linters/CategoryDocstringLinterTest/#CategoryDocstringLinter___not_flagged_test_without_docstring","anchor":"CategoryDocstringLinter___not_flagged_test_without_docstring"},"CategoryDocstringLinter.flagged_textbook_missing_docstring":{"url":"/FormalConjectures/Util/Linters/CategoryDocstringLinterTest/#CategoryDocstringLinter___flagged_textbook_missing_docstring","anchor":"CategoryDocstringLinter___flagged_textbook_missing_docstring"},"CopyrightLinter.correctCopyrightPrefix":{"url":"/FormalConjectures/Util/Linters/CopyrightLinter/#CopyrightLinter___correctCopyrightPrefix","anchor":"CopyrightLinter___correctCopyrightPrefix","docHtml":"\n The part of the expected copyright before the year.
"},"CopyrightLinter.correctCopyrightSuffix":{"url":"/FormalConjectures/Util/Linters/CopyrightLinter/#CopyrightLinter___correctCopyrightSuffix","anchor":"CopyrightLinter___correctCopyrightSuffix","docHtml":"\n The part of the expected copyright after the year.
"},"CopyrightLinter.hasCorrectCopyright":{"url":"/FormalConjectures/Util/Linters/CopyrightLinter/#CopyrightLinter___hasCorrectCopyright","anchor":"CopyrightLinter___hasCorrectCopyright","docHtml":"\n Check whether a file, given as a String, is prefixed with the correct copyright header.
\n The current correct copyright header.
"},"CopyrightLinter.linter.style.copyright.formalConjectures":{"url":"/FormalConjectures/Util/Linters/CopyrightLinter/#CopyrightLinter___linter___style___copyright___formalConjectures","anchor":"CopyrightLinter___linter___style___copyright___formalConjectures"},"CopyrightLinter.copyrightLinter":{"url":"/FormalConjectures/Util/Linters/CopyrightLinter/#CopyrightLinter___copyrightLinter","anchor":"CopyrightLinter___copyrightLinter","docHtml":"\n The copyright linter ensures that every file has the right copyright header.
"},"linter.style.moduleDocstring":{"url":"/FormalConjectures/Util/Linters/ModuleDocstringLinter/#linter___style___moduleDocstring","anchor":"linter___style___moduleDocstring","docHtml":"\n This file implements a linter that enforces module docstring hygiene:
\n\nMissing module docstring: warns when a file has no /-! ... -/\nblock at all (detected on the first non-moduleDoc command).
\nDuplicate module docstrings: warns when a file has more than one\n/-! ... -/ block (subsequent blocks should be regular /- ... -/\ncomments).
\n Files whose first non-moduleDoc command has already been\nprocessed (so we emit the \"missing\" warning at most once).
\n The module docstring linter checks two things:
\n\n A file that has no /-! ... -/ block gets a warning on its\nfirst non-moduleDoc command.
\n A file that has more than one /-! ... -/ block gets a\nwarning on each extra block.
\n The AnswerLinter is a linter to aid with using answer(sorry) correctly.
\n The actual linter object
"},"linter.style.category_docstring":{"url":"/FormalConjectures/Util/Linters/CategoryDocstringLinter/#linter___style___category_docstring","anchor":"linter___style___category_docstring","docHtml":"\n The CategoryDocstringLinter ensures that declarations tagged as\n@[category research open], @[category research solved], or @[category textbook] have a docstring.
\n Extract the category attributes from a declaration's modifiers.
\n Extract the categories from a declaration's modifiers.
"},"CategoryDocstringLinter.categoryNeedsDocstring":{"url":"/FormalConjectures/Util/Linters/CategoryDocstringLinter/#CategoryDocstringLinter___categoryNeedsDocstring","anchor":"CategoryDocstringLinter___categoryNeedsDocstring","docHtml":"\n Whether the given category requires a docstring.
"},"CategoryDocstringLinter.hasDocstring":{"url":"/FormalConjectures/Util/Linters/CategoryDocstringLinter/#CategoryDocstringLinter___hasDocstring","anchor":"CategoryDocstringLinter___hasDocstring","docHtml":"\n Whether the declaration modifiers contain a docstring.
"},"CategoryDocstringLinter.categoryDocstringLinter":{"url":"/FormalConjectures/Util/Linters/CategoryDocstringLinter/#CategoryDocstringLinter___categoryDocstringLinter","anchor":"CategoryDocstringLinter___categoryDocstringLinter","docHtml":"\n The linter checking for docstrings on research statements.
"},"linter.style.category_attribute":{"url":"/FormalConjectures/Util/Linters/CategoryLinter/#linter___style___category_attribute","anchor":"linter___style___category_attribute","docHtml":"\n The categoryLinter is a linter to aid with formatting contributions to\nthe Formal Conjectures repository by ensuring that results in a file have\nthe appropriate tags in order to distinguish between open/already solved\nproblems and background results/sanity checks.
\n Checks if a command has the category attribute.
\n The problem category linter checks that every theorem/lemma/example\nhas been given a problem category attribute.
"},"NamespaceLinter.linter.style.namespace":{"url":"/FormalConjectures/Util/Linters/NamespaceLinter/#NamespaceLinter___linter___style___namespace","anchor":"NamespaceLinter___linter___style___namespace","docHtml":"\n The namespace linter is set on by default. It emits a warning on any declaration\nthat is not inside a namespace (i.e., declarations at the root level).
\n For instance, theorem foo would trigger a warning, while Nat.foo would not.
\n This helps maintain organization and avoid polluting the global namespace.
"},"NamespaceLinter.namespaceLinter":{"url":"/FormalConjectures/Util/Linters/NamespaceLinter/#NamespaceLinter___namespaceLinter","anchor":"NamespaceLinter___namespaceLinter","docHtml":"\n The namespace linter checks that every declaration is inside a namespace.
"},"ExistsImplicationLinter.linter.style.existsImplication":{"url":"/FormalConjectures/Util/Linters/ExistsImplicationLinter/#ExistsImplicationLinter___linter___style___existsImplication","anchor":"ExistsImplicationLinter___linter___style___existsImplication","docHtml":"\n Many misformalisations stem from using a pattern of the form ∃ x, P x → Q instead of\n∃ x, P x ∧ Q (e.g. when formalising something of the form \"there is positive x such that ...\").\nThis is almost always incorrect (and trivial to prove) since it then suffices to pick an x that\ndoes not satisfy P. This linter flags occurences of this patter to the user and proposes a\ncorrected syntax.
\n Changes and expression of the form ∀ (h1 : Prop1) (h2 : Prop2) ..., Propn to\nProp1 ∧ Prop2 ∧ ... ∧ Propn.
\n Checks if an expression contains the pattern ∃ x, P x → Q.
\n The existsImplicationLinter detects expressions of the form ∃ a, P a → Q and flags them to the\nuser since those are rarely correct.
\n This name is here due to the reappearance of https://github.com/leanprover/lean4/issues/10175.
"},"AnswerLinter.flagged_by_linter":{"url":"/FormalConjectures/Util/Linters/AnswerLinterTest/#AnswerLinter___flagged_by_linter","anchor":"AnswerLinter___flagged_by_linter","docHtml":"\n An exampe of what we want to lint against
"},"AnswerLinter.not_flagged_no_answer_sorry":{"url":"/FormalConjectures/Util/Linters/AnswerLinterTest/#AnswerLinter___not_flagged_no_answer_sorry","anchor":"AnswerLinter___not_flagged_no_answer_sorry","docHtml":"\n An non-exampe: we want don't to lint against this case
"},"AnswerLinter.not_flagged_no_arguments":{"url":"/FormalConjectures/Util/Linters/AnswerLinterTest/#AnswerLinter___not_flagged_no_arguments","anchor":"AnswerLinter___not_flagged_no_arguments","docHtml":"\n An non-exampe: we want don't to lint against this case
"},"AnswerLinter.not_flagged_non_prop_answer":{"url":"/FormalConjectures/Util/Linters/AnswerLinterTest/#AnswerLinter___not_flagged_non_prop_answer","anchor":"AnswerLinter___not_flagged_non_prop_answer","docHtml":"\n An non-exampe: here answer(sorry) is not a Prop, and not the entire left\nhand side of the iff.
\n A type that captures the current setting for the answer() elaborator.
\n Default mode: answer(sorry) defaults to True when sorry has type Prop.
\n Default mode for answer(foo): just postpones elaboration.
\n Elaborate answer(foo) by creating an auxiliary definition with value foo.
\n Find the first subexpression carrying the answer annotation,\nreturning the inner (unwrapped) expression if found.
\n Collect answer annotation,\nreturning the inner (unwrapped) expressions.
\n Collect answer annotation,\nreturning the inner (unwrapped) expressions.
\n Indicates where the answer is in a problem statement.
"},"Google.AnswerInfo":{"url":"/FormalConjectures/Util/Answer/#Google___AnswerInfo","anchor":"Google___AnswerInfo","docHtml":"\n An answer: a term, and the context in which it was elaborated
"},"Google.AnswerInfo.ctx":{"url":"/FormalConjectures/Util/Answer/#Google___AnswerInfo___ctx","anchor":"Google___AnswerInfo___ctx","docHtml":"\n An answer: a term, and the context in which it was elaborated
"},"Google.AnswerInfo.term":{"url":"/FormalConjectures/Util/Answer/#Google___AnswerInfo___term","anchor":"Google___AnswerInfo___term","docHtml":"\n An answer: a term, and the context in which it was elaborated
"},"Google.AnswerInfo.format":{"url":"/FormalConjectures/Util/Answer/#Google___AnswerInfo___format","anchor":"Google___AnswerInfo___format","docHtml":"\n Print an answer
"},"Google.getAnswers":{"url":"/FormalConjectures/Util/Answer/#Google___getAnswers","anchor":"Google___getAnswers","docHtml":"\n Find answers by inspecting an InfoTree
\n Find answers by inspecting an InfoTree
\n A topological group G admits a Lie group structure if there exists a finite-dimensional\nsmooth manifold structure on G making it a real Lie group.
\n Every Lie group trivially admits a Lie group structure.
"},"Hilbert5.locallyCompact_of_admitsLieGroupStructure":{"url":"/FormalConjectures/HilbertProblems/«5»/#Hilbert5___locallyCompact_of_admitsLieGroupStructure","anchor":"Hilbert5___locallyCompact_of_admitsLieGroupStructure","docHtml":"\n A group admitting a Lie group structure is locally compact.
"},"Hilbert5.hilbert_smith_conjecture":{"url":"/FormalConjectures/HilbertProblems/«5»/#Hilbert5___hilbert_smith_conjecture","anchor":"Hilbert5___hilbert_smith_conjecture","docHtml":"\nHilbert–Smith conjecture: every locally compact topological group acting continuously\nand faithfully on a connected finite-dimensional topological manifold is a Lie group.
"},"Hilbert5.hilbert_smith_conjecture.variants.riemannian":{"url":"/FormalConjectures/HilbertProblems/«5»/#Hilbert5___hilbert_smith_conjecture___variants___riemannian","anchor":"Hilbert5___hilbert_smith_conjecture___variants___riemannian","docHtml":"\n The conjecture holds when G acts by isometries on a Riemannian manifold, since G\nembeds as a closed subgroup of the isometry group, which is a Lie group by Myers–Steenrod.
\n Pardon (2013): the Hilbert–Smith conjecture holds for 3-dimensional manifolds.\nSee arXiv:1112.2324.
"},"Hilbert5.hilbert_smith_padic_formulation":{"url":"/FormalConjectures/HilbertProblems/«5»/#Hilbert5___hilbert_smith_padic_formulation","anchor":"Hilbert5___hilbert_smith_padic_formulation","docHtml":"\n Equivalent p-adic formulation: the p-adic integers ℤ_[p] cannot act continuously and\nfaithfully on any connected finite-dimensional topological manifold. By the Gleason–Yamabe\ntheorem, this is equivalent to hilbert_smith_conjecture.
\nHilbert's fifth problem (Gleason–Montgomery–Zippin, 1952): every locally Euclidean\ntopological group is a Lie group.
"},"Hilbert17.hilbert_17th_problem":{"url":"/FormalConjectures/HilbertProblems/«17»/#Hilbert17___hilbert_17th_problem","anchor":"Hilbert17___hilbert_17th_problem","docHtml":"\n Hilbert's 17th problem: every non-negative multivariate polynomial is a sum of\nsquares of rational functions.
"},"Hilbert17.f":{"url":"/FormalConjectures/HilbertProblems/«17»/#Hilbert17___f","anchor":"Hilbert17___f","docHtml":"\n The statement is false in general if we restrict to polynomials. The polynomial (by Motzkin)\n$f(x, y) = x^4 y^2 + x^2 y^4 - 3 x^2 y^2 + 1$ takes only nonnegative values but cannot be\nwritten as a sum of squares of polynomials.
"},"Hilbert17.f_nonneg":{"url":"/FormalConjectures/HilbertProblems/«17»/#Hilbert17___f_nonneg","anchor":"Hilbert17___f_nonneg","docHtml":"\n The Motzkin polynomial is non-negative everywhere.
"},"Hilbert17.f_not_sum_of_squares":{"url":"/FormalConjectures/HilbertProblems/«17»/#Hilbert17___f_not_sum_of_squares","anchor":"Hilbert17___f_not_sum_of_squares","docHtml":"\n The Motzkin polynomial cannot be written as a sum of squares of polynomials.
"},"Hilbert17.Hilbert17thProblemHomogenousPoly":{"url":"/FormalConjectures/HilbertProblems/«17»/#Hilbert17___Hilbert17thProblemHomogenousPoly","anchor":"Hilbert17___Hilbert17thProblemHomogenousPoly","docHtml":"\n For the polynomial version, Hilbert showed that every nonnegative homogeneous polynomial in\n$n$ variables of degree $2d$ can be written as a sum of squares of polynomials if and only if
\n\n $n = 1$
\n\n $n = 2$
\n\n $d = 1$
\n\n $(n, d) = (3, 2)$.
\n\n Hilbert's 17th problem for homogeneous polynomials: characterization of dimensions and degrees\nwhere non-negative polynomials are sums of squares of polynomials.
"},"Arxiv.«math.0110202».banach_mazur_rotation_problem":{"url":"/FormalConjectures/Arxiv/«math.0110202»/BanachMazurRotation/#Arxiv____FLQQ_math___0110202_FLQQ____banach_mazur_rotation_problem","anchor":"Arxiv____FLQQ_math___0110202_FLQQ____banach_mazur_rotation_problem","docHtml":"\n The Banach--Mazur rotation problem asks whether every separable Banach space whose group of linear\nisometric equivalences acts transitively on the unit sphere is linearly isometric to a Hilbert\nspace.
"},"Arxiv.«math.0110202».banach_mazur_rotation_problem.finite_dimensional":{"url":"/FormalConjectures/Arxiv/«math.0110202»/BanachMazurRotation/#Arxiv____FLQQ_math___0110202_FLQQ____banach_mazur_rotation_problem___finite_dimensional","anchor":"Arxiv____FLQQ_math___0110202_FLQQ____banach_mazur_rotation_problem___finite_dimensional","docHtml":"\n Every finite-dimensional real normed space whose isometry group acts transitively on the\nunit sphere is Euclidean.
"},"Arxiv.«1601.03081».IsCrystalWithComponents":{"url":"/FormalConjectures/Arxiv/«1601.03081»/UniqueCrystalComponents/#Arxiv____FLQQ_1601___03081_FLQQ____IsCrystalWithComponents","anchor":"Arxiv____FLQQ_1601___03081_FLQQ____IsCrystalWithComponents","docHtml":"\n An odd number $n$ is called a crystal if $n = ab$, with $a, b > 1$\nand $B(a, b) ∈ ℕ$, where $B(a, b) := ((a + b)^2 + (a b + 1)^2) / (2 (a + 1) (b + 1))$.
"},"Arxiv.«1601.03081».isCrystalWithComponents_35_5_7":{"url":"/FormalConjectures/Arxiv/«1601.03081»/UniqueCrystalComponents/#Arxiv____FLQQ_1601___03081_FLQQ____isCrystalWithComponents_35_5_7","anchor":"Arxiv____FLQQ_1601___03081_FLQQ____isCrystalWithComponents_35_5_7"},"Arxiv.«1601.03081».crystals_components_unique":{"url":"/FormalConjectures/Arxiv/«1601.03081»/UniqueCrystalComponents/#Arxiv____FLQQ_1601___03081_FLQQ____crystals_components_unique","anchor":"Arxiv____FLQQ_1601___03081_FLQQ____crystals_components_unique","docHtml":"\n If $n = ab$ is a crystal, then there are no other pairs of\npositive integers $c, d > 1$, different from the couple $a, b$, such that $n = cd$ and\n$B(c, d) ∈ ℕ$, i.e., the components of the crystals are unique.
"},"Arxiv.id2303_01089.MultiplicativelyIndependent":{"url":"/FormalConjectures/Arxiv/«2303.01089»/FurstenbergTimesPTimesQ/#Arxiv___id2303_01089___MultiplicativelyIndependent","anchor":"Arxiv___id2303_01089___MultiplicativelyIndependent","docHtml":"\n Two integers $p, q \\ge 2$ are multiplicatively independent if\n$\\log p / \\log q$ is irrational.
"},"Arxiv.id2303_01089.Tn":{"url":"/FormalConjectures/Arxiv/«2303.01089»/FurstenbergTimesPTimesQ/#Arxiv___id2303_01089___Tn","anchor":"Arxiv___id2303_01089___Tn","docHtml":"\n The map $T_n$ sends $x$ to $nx \\bmod 1$ on the additive circle.
"},"Arxiv.id2303_01089.Tn_continuous":{"url":"/FormalConjectures/Arxiv/«2303.01089»/FurstenbergTimesPTimesQ/#Arxiv___id2303_01089___Tn_continuous","anchor":"Arxiv___id2303_01089___Tn_continuous"},"Arxiv.id2303_01089.IsTnInvariant":{"url":"/FormalConjectures/Arxiv/«2303.01089»/FurstenbergTimesPTimesQ/#Arxiv___id2303_01089___IsTnInvariant","anchor":"Arxiv___id2303_01089___IsTnInvariant","docHtml":"\n A set $F$ is $T_n$-invariant if $T_n(F) \\subseteq F$.
"},"Arxiv.id2303_01089.MeasureTheory.IsAtom":{"url":"/FormalConjectures/Arxiv/«2303.01089»/FurstenbergTimesPTimesQ/#Arxiv___id2303_01089___MeasureTheory___IsAtom","anchor":"Arxiv___id2303_01089___MeasureTheory___IsAtom","docHtml":"\n A set $A$ is an atom if it has positive measure and for all $B \\subseteq A$ measurable,\neither $\\mu(B) = 0$ or $\\mu(B) = \\mu(A)$.
"},"Arxiv.id2303_01089.MeasureTheory.IsAtomLess":{"url":"/FormalConjectures/Arxiv/«2303.01089»/FurstenbergTimesPTimesQ/#Arxiv___id2303_01089___MeasureTheory___IsAtomLess","anchor":"Arxiv___id2303_01089___MeasureTheory___IsAtomLess","docHtml":"\n A measure is atomless if it has no atoms.
"},"Arxiv.id2303_01089.MeasureTheory.IsAtomLess.NoAtoms":{"url":"/FormalConjectures/Arxiv/«2303.01089»/FurstenbergTimesPTimesQ/#Arxiv___id2303_01089___MeasureTheory___IsAtomLess___NoAtoms","anchor":"Arxiv___id2303_01089___MeasureTheory___IsAtomLess___NoAtoms","docHtml":"\n A measure is atomless if it has no atoms.
"},"Arxiv.id2303_01089.UnitAddCircle.ProbabilityMeasure":{"url":"/FormalConjectures/Arxiv/«2303.01089»/FurstenbergTimesPTimesQ/#Arxiv___id2303_01089___UnitAddCircle___ProbabilityMeasure","anchor":"Arxiv___id2303_01089___UnitAddCircle___ProbabilityMeasure"},"Arxiv.id2303_01089.conjecture_1_3":{"url":"/FormalConjectures/Arxiv/«2303.01089»/FurstenbergTimesPTimesQ/#Arxiv___id2303_01089___conjecture_1_3","anchor":"Arxiv___id2303_01089___conjecture_1_3","docHtml":"\nConjecture 1.3 (the $\\times p, \\times q$ conjecture): the only atomless Borel probability\nmeasure on $\\mathbb{T}$ which is both $T_p$- and $T_q$-invariant is the Lebesgue measure.
"},"Arxiv.id2303_01089.conjecture_1_4":{"url":"/FormalConjectures/Arxiv/«2303.01089»/FurstenbergTimesPTimesQ/#Arxiv___id2303_01089___conjecture_1_4","anchor":"Arxiv___id2303_01089___conjecture_1_4","docHtml":"\nConjecture 1.4: if $\\mu$ is an atomless $T_p$-invariant Borel probability measure on\n$\\mathbb{T}$, then $T_{q^n}\\mu$ converges weak-star to Lebesgue measure.\nThis paper disproves the conjecture.
"},"Arxiv.«2501.03234».S'":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____S___","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____S___","docHtml":"\n Define the sum\n$$S'(h, k) := \\sum_{j=1}^{k-1}(-1)^{j + 1 + \\lfloor \\frac{hj}{k}\\rfloor}.$$
"},"Arxiv.«2501.03234».S":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____S","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____S","docHtml":"\n Define the sum\n$$S(k) := \\sum_{h=1}^{k-1}S'(h, k)$$
"},"Arxiv.«2501.03234».S_fst_10":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____S_fst_10","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____S_fst_10","docHtml":"\n Note that in Table 1 in https://arxiv.org/abs/2501.03234v1, there seems to be an error:\n11 appears twice. The first 10 values of $S$.
"},"Arxiv.«2501.03234».conjecture_1_1":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____conjecture_1_1","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____conjecture_1_1","docHtml":"\nConjecture 1.1: For any odd prime $k$, the sum associated with the classical theta function $θ_3$,\n$S(k)$ is positive.
"},"Arxiv.«2501.03234».conjecture_4_1":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_1","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_1","docHtml":"\nConjecture 4.1: For any prime $k$ larger than $5$, $S(k) > k$.
"},"Arxiv.«2501.03234».conjecture_4_2":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_2","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_2","docHtml":"\nConjecture 4.2: For any prime $k$ larger than $233$, $S(k) > 2k$.
"},"Arxiv.«2501.03234».conjecture_4_3":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_3","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_3","docHtml":"\nConjecture 4.3: For any prime $k$ larger than $3119$, $S(k) > 3k$.
"},"Arxiv.«2501.03234».conjecture_4_4":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_4","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_4","docHtml":"\nConjecture 4.4: Given a natural number $n ∈ ℕ$, for all large enough odd prime $k$ (depending on $n$),\n$nk < S(k)$.
"},"Arxiv.«2501.03234».conjecture_4_4_def_0":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_4_def_0","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_4_def_0","docHtml":"\nConjecture 1.1 → Conjecture 4.4: If conjecture 1.1 holds true, then this implies a special\ncase of conjecture 4.4 where $n = 0$. In this case the lower bound for the odd prime $k$\nwould be $0$.
"},"Arxiv.«2501.03234».conjecture_4_4_def_1":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_4_def_1","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_4_def_1","docHtml":"\nConjecture 4.1 → Conjecture 4.4: If conjecture 4.1 holds true, then this implies a special\ncase of conjecture 4.4 where $n = 1$. In this case the lower bound would be $5$.
"},"Arxiv.«2501.03234».conjecture_4_4_def_2":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_4_def_2","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_4_def_2","docHtml":"\nConjecture 4.2 → Conjecture 4.4: If conjecture 4.2 holds true, then this implies a special\ncase of conjecture 4.4 for $n = 2$. For this scenario, the lower bound is now $233$.
"},"Arxiv.«2501.03234».conjecture_4_4_def_3":{"url":"/FormalConjectures/Arxiv/«2501.03234»/ArithmeticSumS/#Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_4_def_3","anchor":"Arxiv____FLQQ_2501___03234_FLQQ____conjecture_4_4_def_3","docHtml":"\nConjecture 4.3 → Conjecture 4.4: If conjecture 4.3 holds true, then a special\ncase of conjecture 4.4 for $n = 3$ is obtained, and the lower bound is $3119$.
"},"Margulis.conjecture_1_1":{"url":"/FormalConjectures/Arxiv/«2504.17644»/Margulis/#Margulis___conjecture_1_1","anchor":"Margulis___conjecture_1_1","docHtml":"\n Let D be the diagonal group of SL_n(ℝ) where n ≥ 3.\nThen any relatively compact D-orbit in SL_n(ℝ) / SL_n(ℤ) is closed.
\n The natural inclusion F[t] →+* F((t⁻¹)).
\nHuang–Shi, Theorem 1.2
\n\n Let F be a finite field of characteristic p ∈ {3, 5, 7, 11}, and set\nK = F((t⁻¹)), A = F[t]. Let
\nD be the diagonal subgroup of SL₄(K),
\nΓ = SL₄(A) the lattice subgroup embedded into SL₄(K) via the natural inclusion A →+* K.
\n Then there exists z : SL₄(K)/Γ such that the D-orbit of z has compact\nclosure but is not closed.
\n Let $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ be two triples of integers.\nSay that $a$ is $2$-less than $b$, or $a <_2 b$, if $a_i < b_i$ for at least\ntwo co-ordinates $i$.
"},"Arxiv.«1609.08688».not_lt₂":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____not_lt___","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____not_lt___"},"Arxiv.«1609.08688».not_lt₂_of_forall_le":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____not_lt____of_forall_le","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____not_lt____of_forall_le"},"Arxiv.«1609.08688».not_lt₂_of_exists":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____not_lt____of_exists","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____not_lt____of_exists"},"Arxiv.«1609.08688».not_lt₂_self":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____not_lt____self","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____not_lt____self"},"Arxiv.«1609.08688».lt₂_example_1":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____lt____example_1","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____lt____example_1","docHtml":"\n For example, $(3, 3, 9) <_2 (5, 6, 1)$.
"},"Arxiv.«1609.08688».lt₂_example_2":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____lt____example_2","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____lt____example_2","docHtml":"\n $(5, 6, 1) <_2 (7, 7, 7)$
"},"Arxiv.«1609.08688».lt₂_example_3":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____lt____example_3","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____lt____example_3","docHtml":"\n $(7, 7, 7) <_2 (7, 8, 9)$
"},"Arxiv.«1609.08688».not_lt₂_example":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____not_lt____example","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____not_lt____example","docHtml":"\n but $(1, 2, 3)$ is not $2$-less than $(1, 2, 4).
"},"Arxiv.«1609.08688».not_trans_lt₂_nat":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____not_trans_lt____nat","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____not_trans_lt____nat","docHtml":"\n The $2$-less relation is not transitive on the naturals.
"},"Arxiv.«1609.08688».IsIncreasing₂":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____IsIncreasing___","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____IsIncreasing___","docHtml":"\n Since the $2$-less relation is not transitive, we make a further definition to\nspecify transivity.
"},"Arxiv.«1609.08688».isIncreasing₂_nil":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____isIncreasing____nil","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____isIncreasing____nil"},"Arxiv.«1609.08688».isIncreasing₂_singleton":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____isIncreasing____singleton","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____isIncreasing____singleton"},"Arxiv.«1609.08688».isIncreasing₂_const_length":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____isIncreasing____const_length","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____isIncreasing____const_length"},"Arxiv.«1609.08688».maximalLength":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____maximalLength","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____maximalLength","docHtml":"\n Let $F(n)$ be the maximal length of a $2$-increasing sequence of triples with each coordinate\nbelong to $[n]$ ($= {1, 2, ..., n}$).
"},"Arxiv.«1609.08688».maximalLength_zero":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_zero","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_zero"},"Arxiv.«1609.08688».maximalLength_one":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_one","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_one"},"Arxiv.«1609.08688».maximalLength_four":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_four","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_four"},"Arxiv.«1609.08688».exists_pair_of_mem_Icc":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____exists_pair_of_mem_Icc","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____exists_pair_of_mem_Icc","docHtml":"\n In a set of more than $n^2$ triples with coordinates from ${1, ..., n}$ we must\nhave two triples that are equal in their first two coordinates.
"},"Arxiv.«1609.08688».maximalLength_le":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_le","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_le","docHtml":"\n For all $n$ we have $F(n) \\leq n^2$.
"},"Arxiv.«1609.08688».maximalLength_ge_of_isSquare":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_ge_of_isSquare","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_ge_of_isSquare","docHtml":"\n Moreover, whenever $n$ is a perfect square we have $F(n) \\geq n^{3/2}$.
"},"Arxiv.«1609.08688».IsComparable₂":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____IsComparable___","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____IsComparable___","docHtml":"\n Two triples $t_1$ and $t_2$ are $2$-comparable if one of them is $2$-less\nthan the other.
"},"Arxiv.«1609.08688».IsComparableSet₂":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____IsComparableSet___","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____IsComparableSet___","docHtml":"\n A set of triples is $2$-comparable if any two of them are $2$-comparable.
"},"Arxiv.«1609.08688».maximalLength_le_isBigO":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_le_isBigO","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_le_isBigO","docHtml":"\n $F(n) \\leq n^2 / \\exp(\\Omega(\\log^*(n)))$.
"},"Arxiv.«1609.08688».tripleProduct":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____tripleProduct","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____tripleProduct","docHtml":"\n We define the product of two triples $(a, b, c)$ and $(d, e, f)$ by\n$((a, d), (b, e), (c, f))$, where the pairs are arranged in lexicographical order.
"},"Arxiv.«1609.08688».tripleProduct_const":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____tripleProduct_const","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____tripleProduct_const"},"Arxiv.«1609.08688».tripleProduct_vecConst_const":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____tripleProduct_vecConst_const","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____tripleProduct_vecConst_const"},"Arxiv.«1609.08688».sequenceProduct":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____sequenceProduct","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____sequenceProduct","docHtml":"\n We define the product $\\otimes$ of two sequences $(a_i, b_i, c_i)$ and\n$(d_i, e_i, f_i)$ by the sequence $((a_i, d_j), (b_i, e_j), (c_i, f_j))$, where\nthe indices $(i, j)$ are arranged lexicographically, and the pairs are also\nordered lexicographically.
"},"Arxiv.«1609.08688».sequenceProduct_example":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____sequenceProduct_example","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____sequenceProduct_example"},"Arxiv.«1609.08688».maximalLength_pow":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_pow","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_pow","docHtml":"\n Suppose that for some $n$ we have $F(n) = n ^ {\\alpha}$. Then there are arbitrarily\nlarge $m$ such that $F(m) \\geq m^{\\alpha}$.
"},"Arxiv.«1609.08688».maximalLength_le_strong":{"url":"/FormalConjectures/Arxiv/«1609.08688»/sIncreasingrTuples/#Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_le_strong","anchor":"Arxiv____FLQQ_1609___08688_FLQQ____maximalLength_le_strong","docHtml":"\n $F(n) \\leq n^{3/2}$.
"},"Arxiv.«2602.05192».IsEpsilonLight":{"url":"/FormalConjectures/Arxiv/«2602.05192»/FirstProof6/#Arxiv____FLQQ_2602___05192_FLQQ____IsEpsilonLight","anchor":"Arxiv____FLQQ_2602___05192_FLQQ____IsEpsilonLight","docHtml":"\n For a graph $G = (V, E)$, let $G_S = (V, E(S,S))$ denote the graph with the same vertex set,\nbut only the edges between vertices in $S$.\nLet $L$ be the Laplacian matrix of $G$ and let $L_S$ be the Laplacian of $G_S$.
\n\n I say that a set of vertices $S$ is $\\epsilon$-light if the matrix $\\epsilon L - L_S$ is\npositive semidefinite.
"},"Arxiv.«2602.05192».epsilon_light_subset_exists":{"url":"/FormalConjectures/Arxiv/«2602.05192»/FirstProof6/#Arxiv____FLQQ_2602___05192_FLQQ____epsilon_light_subset_exists","anchor":"Arxiv____FLQQ_2602___05192_FLQQ____epsilon_light_subset_exists","docHtml":"\n Does there exist a constant $c > 0$ so that for every graph $G$ and every $\\epsilon$ between\n$0$ and $1$, $V$ contains an $\\epsilon$-light subset $S$ of size at least $c \\epsilon |V|$?
"},"Arxiv.«2602.05192».finiteAdditiveConvolution":{"url":"/FormalConjectures/Arxiv/«2602.05192»/FirstProof4/#Arxiv____FLQQ_2602___05192_FLQQ____finiteAdditiveConvolution","anchor":"Arxiv____FLQQ_2602___05192_FLQQ____finiteAdditiveConvolution","docHtml":"\n Define $p \\boxplus_n q(x)$ to be the polynomial\n$$\n(p \\boxplus_n q)(x) = \\sum_{k=0}^n c_k x^{n-k}\n$$\nwhere the coefficients $c_k$ are given by the formula:\n$$\nc_k = \\sum_{i+j=k} \\frac{(n-i)! (n-j)!}{n! (n-k)!} a_i b_j\n$$\nfor $k = 0, 1, \\dots, n$.
"},"Arxiv.«2602.05192».finiteAdditiveConvolution_comm":{"url":"/FormalConjectures/Arxiv/«2602.05192»/FirstProof4/#Arxiv____FLQQ_2602___05192_FLQQ____finiteAdditiveConvolution_comm","anchor":"Arxiv____FLQQ_2602___05192_FLQQ____finiteAdditiveConvolution_comm"},"Arxiv.«2602.05192».finiteAdditiveConvolution_degree":{"url":"/FormalConjectures/Arxiv/«2602.05192»/FirstProof4/#Arxiv____FLQQ_2602___05192_FLQQ____finiteAdditiveConvolution_degree","anchor":"Arxiv____FLQQ_2602___05192_FLQQ____finiteAdditiveConvolution_degree"},"Arxiv.«2602.05192».finiteAdditiveConvolution_monic'":{"url":"/FormalConjectures/Arxiv/«2602.05192»/FirstProof4/#Arxiv____FLQQ_2602___05192_FLQQ____finiteAdditiveConvolution_monic___","anchor":"Arxiv____FLQQ_2602___05192_FLQQ____finiteAdditiveConvolution_monic___"},"Arxiv.«2602.05192».Φ":{"url":"/FormalConjectures/Arxiv/«2602.05192»/FirstProof4/#Arxiv____FLQQ_2602___05192_FLQQ_______","anchor":"Arxiv____FLQQ_2602___05192_FLQQ_______","docHtml":"\n For a monic polynomial $p(x)=\\prod_{i\\le n}(x- \\lambda_i)$, define\n$$\\Phi_n(p):=\\sum_{i\\le n}(\\sum_{j\\neq i} \\frac1{\\lambda_i-\\lambda_j})^2$$\nand $\\Phi_n(p):=\\infty$ if $p$ has a multiple root.
"},"Arxiv.«2602.05192».FourProp":{"url":"/FormalConjectures/Arxiv/«2602.05192»/FirstProof4/#Arxiv____FLQQ_2602___05192_FLQQ____FourProp","anchor":"Arxiv____FLQQ_2602___05192_FLQQ____FourProp","docHtml":"\n A predicate that holds if $p(x)$ and $q(x)$ are monic real-rooted polynomials of\ndegree $n$, then\n$$\\frac{1}{\\Phi_n(p\\boxplus_n q)} \\ge \\frac{1}{\\Phi_n(p)}+\\frac{1}{\\Phi_n(q)}?$$
"},"Arxiv.«2602.05192».four":{"url":"/FormalConjectures/Arxiv/«2602.05192»/FirstProof4/#Arxiv____FLQQ_2602___05192_FLQQ____four","anchor":"Arxiv____FLQQ_2602___05192_FLQQ____four","docHtml":"\n Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of\ndegree $n$, then\n$$\\frac{1}{\\Phi_n(p\\boxplus_n q)} \\ge \\frac{1}{\\Phi_n(p)}+\\frac{1}{\\Phi_n(q)}?$$
\n\narxiv/2602.05192v2 contains a proof.
"},"Arxiv.«2602.05192».four_2":{"url":"/FormalConjectures/Arxiv/«2602.05192»/FirstProof4/#Arxiv____FLQQ_2602___05192_FLQQ____four_2","anchor":"Arxiv____FLQQ_2602___05192_FLQQ____four_2","docHtml":"\n Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of\ndegree $2$, then\n$$\\frac{1}{\\Phi_2(p\\boxplus_n q)} \\ge \\frac{1}{\\Phi_2(p)}+\\frac{1}{\\Phi_2(q)}?$$
"},"Arxiv.«2602.05192».four_3":{"url":"/FormalConjectures/Arxiv/«2602.05192»/FirstProof4/#Arxiv____FLQQ_2602___05192_FLQQ____four_3","anchor":"Arxiv____FLQQ_2602___05192_FLQQ____four_3","docHtml":"\n Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of\ndegree $3$, then\n$$\\frac{1}{\\Phi_3(p\\boxplus_n q)} \\ge \\frac{1}{\\Phi_3(p)}+\\frac{1}{\\Phi_3(q)}?$$
"},"Arxiv.«2208.14736».IsCancellative":{"url":"/FormalConjectures/Arxiv/«2208.14736»/ZariskiCancellation/#Arxiv____FLQQ_2208___14736_FLQQ____IsCancellative","anchor":"Arxiv____FLQQ_2208___14736_FLQQ____IsCancellative","docHtml":"\n A finitely generated k-algebra A is cancellative if for all finitely generated k algebras B such that\nB[X] ≅ₖ A[X] we have B ≅ₖ A.
\n The Zariski Cancellation Problem: every polynomial ring over a field k of characteristic\n0 is cancellative.
\n The single variable polynomial ring k[X] is cancellative in any characteristic
\n The two variable polynomial ring k[X] is cancellative in any characteristic
\n The positive characteristic case of the Zariski Cancellation Problem is false in dimension 3
\n For $n > 8$, $2^n$ is not the the sum of distinct powers of $3$. Expressed here in terms of the base $3$ digits of $n$.
\n\n This conjecture is equivalent to the halting of a $15$-state $2$-symbol Turing Machine.
\n\n TODO(lezeau): Formalize the Turing Machine version of this problem.
\n\n Source:
\n For $n = 8$, $2$ is not contained in the base $3$ digits of $n$.
"},"Arxiv.«0912.2382».k":{"url":"/FormalConjectures/Arxiv/«0912.2382»/CurlingNumberConjecture/#Arxiv____FLQQ_0912___2382_FLQQ____k","anchor":"Arxiv____FLQQ_0912___2382_FLQQ____k","docHtml":"\n The curling number
\n\n Let $S$ be a finite nonempty sequence of integers. By grouping adjacent terms, it is always possible\nto write it as $S = X Y Y . . . Y = X Y^k$, where $X$ and $Y$ are sequences of integers and $Y$ is nonempty\n($X$ is allowed to be the empty sequence $∅$). There may be several ways to do this: choose the one\nthat maximizes the value of $k$: this $k$ is the curling number of $S$, denoted by $k S$.
"},"Arxiv.«0912.2382».S":{"url":"/FormalConjectures/Arxiv/«0912.2382»/CurlingNumberConjecture/#Arxiv____FLQQ_0912___2382_FLQQ____S","anchor":"Arxiv____FLQQ_0912___2382_FLQQ____S","docHtml":"\n One starts with any initial\nsequence of integers $S₀$, and extends it by repeatedly appending the curling number of the current\nsequence.
"},"Arxiv.«0912.2382».curling_number_conjecture":{"url":"/FormalConjectures/Arxiv/«0912.2382»/CurlingNumberConjecture/#Arxiv____FLQQ_0912___2382_FLQQ____curling_number_conjecture","anchor":"Arxiv____FLQQ_0912___2382_FLQQ____curling_number_conjecture","docHtml":"\n The sequence will eventually reach $1$.
"},"Arxiv.«2107.00295».independentDominationEven":{"url":"/FormalConjectures/Arxiv/«2107.00295»/IndependentDomination/#Arxiv____FLQQ_2107___00295_FLQQ____independentDominationEven","anchor":"Arxiv____FLQQ_2107___00295_FLQQ____independentDominationEven","docHtml":"\nConjecture 1.6 (Even case).\nFor a nonempty isolate-free graph $G$ on $n$ vertices,\nif $D$ is even, then $(D + 2)^2 \\cdot i(G) \\leq (D^2 + 4) \\cdot n$.
"},"Arxiv.«2107.00295».independentDominationOdd":{"url":"/FormalConjectures/Arxiv/«2107.00295»/IndependentDomination/#Arxiv____FLQQ_2107___00295_FLQQ____independentDominationOdd","anchor":"Arxiv____FLQQ_2107___00295_FLQQ____independentDominationOdd","docHtml":"\nConjecture 1.6 (Odd case).\nFor a nonempty isolate-free graph $G$ on $n$ vertices,\nif $D$ is odd, then $(D + 1)(D + 3) \\cdot i(G) \\leq (D^2 + 3) \\cdot n$.
"},"Arxiv.«0911.2077».arxiv.id0911_2077.conjecture6_3":{"url":"/FormalConjectures/Arxiv/«0911.2077»/Conjecture6_3/#Arxiv____FLQQ_0911___2077_FLQQ____arxiv___id0911_2077___conjecture6_3","anchor":"Arxiv____FLQQ_0911___2077_FLQQ____arxiv___id0911_2077___conjecture6_3","docHtml":"\n Empirical evidence seems to suggest that Slud's bound does not hold for all $p$, and in fact, as $n\\to\\infty$,\nthe maximal permissible $p$ shrinks to $\\frac{1}{2}$. Also, the following appears to be true:
\n\n When $p\\in(0,1/2)$ and\n$m = 2k$ is even, and $\\sigma := \\sqrt{p(1-p)}$,\n$$\n\\mathbb{P}[B(p,m) \\geq m/2] \\geq 1 - \\Phi\\left(\\frac{(1/2-p)\\sqrt{m}}{\\sigma}\\right) + \\frac 1 2\\binom{m}{m/2}\\sigma^{m}.\n$$
\n\n A solution of this statement has been put out by Logical Intelligence\nhttps://github.com/logical-intelligence/proofs, see\nhere for\nand informal sketch of the proof.
"},"Arxiv.«1308.0994».Formula":{"url":"/FormalConjectures/Arxiv/«1308.0994»/BoxdotConjecture/#Arxiv____FLQQ_1308___0994_FLQQ____Formula","anchor":"Arxiv____FLQQ_1308___0994_FLQQ____Formula","docHtml":"\nFormula is the inductive type of propositional modal formulas:
\nAtom n is a propositional variable indexed by n.
\nFalsum is the constant ⊥.
\nImp α β is implication (α → β).
\nNec α is the necessity operator □α.
\nAtom n is a propositional variable indexed by n.
\nFalsum is the constant ⊥.
\nImp α β is implication (α → β).
\nNec α is the necessity operator □α.
\nConj α β is the conjunction α ∧ β. We define α & β as ~(α ~> ~β) for simplicity.
\nt φ is the Boxdot translation of a formula φ. Roughly, t is the mapping φ ↦ t φ\nfrom the language of monomodal logic into itself that preserves variables and the logical constant ⊥,\ncommutes with the standard truth-functional operators, and is such that t □a = □t a & t a.\nThis implementation follows the definition in Steinsvold (AJL).
\nKProof Γ φ is the usual Hilbert‐style proof relation for the minimal normal modal logic K,\nwith assumptions drawn from Γ.
\n Assumption rule: if α ∈ Γ then α is provable from Γ.
\n Ax1: every instance of the schema α → (β → α) is a theorem.
\n Ax2: every instance of the schema (α ~> β ~> γ) ~> (α ~> β) ~> (α ~> γ) is a theorem.
\n Ax3 (contraposition): every instance of the schema (~α ~> ~β) ~> (β ~> α) is a theorem.
\n Modus Ponens: if Γ ⊢ α ~> β and Γ ⊢ α, then Γ ⊢ β.
\n Necessitation: if ⊢ α then ⊢ □α.
\n Distribution: every instance of the schema □(α ~> β) ~> (□α ~> □β) is a theorem.
\nKTProof Γ φ denotes that φ is provable from the premises Γ in the normal modal logic KT\n(also called T). KT extends system K by adding the instances of the T-axiom schema □φ ~> φ to K’s\nusual axioms and rules of inference.
\n Embedding of K proofs into KT.
"},"Arxiv.«1308.0994».KTProof.axT":{"url":"/FormalConjectures/Arxiv/«1308.0994»/BoxdotConjecture/#Arxiv____FLQQ_1308___0994_FLQQ____KTProof___axT","anchor":"Arxiv____FLQQ_1308___0994_FLQQ____KTProof___axT","docHtml":"\n T-axiom schema: every instance of □α ~> α is a theorem.
\n Modus Ponens: if Γ ⊢ α ~> β and Γ ⊢ α, then Γ ⊢ β.
\n Necessitation: if ⊢ α then ⊢ □α.
\n If KProof Γ φ, then KTProof Γ φ. In other words, KT extends K.
\n A “normal modal logic” L is any Set Formula such that:
\n If K ⊢ φ, then φ ∈ L (L extends K)
\n If φ ∈ L and (φ ~> ψ) ∈ L, then ψ ∈ L (Closed under MP)
\n If φ ∈ L, then □φ ∈ L (Closed under Necessitation)
\nthms is the set of formulas proveable in the logic.
\nextK means that if K ⊢ φ, then φ ∈ thms. That is, the logic extends system K.
\nmp means that if φ ∈ thms and (φ ~> ψ) ∈ thms, then ψ ∈ thms. That is, thms is closed\nunder modus ponens.
\nnec means that if φ ∈ thms, then □φ ∈ thms. Equivalently, thms is closed under\nnecessitation
\nKT is the specific normal modal logic whose theorems are exactly those provable\nin KTProof from the empty context.
\n This corresponds to K ⊕ (□φ → φ) as in both AJL (Steinsvold) and Jeřábek.
\n Boxdot Conjecture: every normal modal logic that faithfully interprets KT\nby the boxdot translation is included in KT.
"},"ComplexityTheory.DecisionProblem":{"url":"/FormalConjectures/Millenium/PvsNP/#ComplexityTheory___DecisionProblem","anchor":"ComplexityTheory___DecisionProblem","docHtml":"\n The type of decision problems.
\n\n We define these as functions from lists of booleans to booleans,\nimplictly assuming the usual encodings.
"},"ComplexityTheory.ComplexityClass":{"url":"/FormalConjectures/Millenium/PvsNP/#ComplexityTheory___ComplexityClass","anchor":"ComplexityTheory___ComplexityClass","docHtml":"\n The type of complexity classes. We define these as sets of decision problems.
"},"ComplexityTheory.IsComputableInPolyTime":{"url":"/FormalConjectures/Millenium/PvsNP/#ComplexityTheory___IsComputableInPolyTime","anchor":"ComplexityTheory___IsComputableInPolyTime","docHtml":"\n A simple definition to abstract the notion of a poly-time Turing machine into a predicate.
"},"ComplexityTheory.P":{"url":"/FormalConjectures/Millenium/PvsNP/#ComplexityTheory___P","anchor":"ComplexityTheory___P","docHtml":"\n The class P is the set of decision problems\ndecidable in polynomial time by a deterministic Turing machine.
"},"ComplexityTheory.NP":{"url":"/FormalConjectures/Millenium/PvsNP/#ComplexityTheory___NP","anchor":"ComplexityTheory___NP","docHtml":"\n The class NP is the set of decision problems\nsuch that there exists a polynomial p over ℕ and a poly-time Turing machine\nwhere for all x, L x = true iff there exists a w of length at most p (|x|)\nsuch that the Turing machine accepts the pair (x,w).
\n See Definition 2.1 in Arora-Barak (2009).
"},"ComplexityTheory.coNP":{"url":"/FormalConjectures/Millenium/PvsNP/#ComplexityTheory___coNP","anchor":"ComplexityTheory___coNP","docHtml":"\n The class coNP is the set of decision problems\nwhose complements are in NP.
"},"ComplexityTheory.P_ne_NP":{"url":"/FormalConjectures/Millenium/PvsNP/#ComplexityTheory___P_ne_NP","anchor":"ComplexityTheory___P_ne_NP","docHtml":"\nP ≠ NP:
\n\n The conjecture that the complexity classes P and NP are not equal.
"},"ComplexityTheory.NP_ne_coNP":{"url":"/FormalConjectures/Millenium/PvsNP/#ComplexityTheory___NP_ne_coNP","anchor":"ComplexityTheory___NP_ne_coNP","docHtml":"\nNP ≠ coNP:
\n\n The conjecture that the complexity classes NP and coNP are not equal.
"},"ComplexityTheory.coP_eq_P":{"url":"/FormalConjectures/Millenium/PvsNP/#ComplexityTheory___coP_eq_P","anchor":"ComplexityTheory___coP_eq_P","docHtml":"\n The theorem that the set of complements of languages in P is itself P.
\n\n This can be proven by observing that the boolean negation function is computable in polynomial time,\nand that compositions of poly-time computable functions are also poly-time computable.
"},"ComplexityTheory.P_subset_NP":{"url":"/FormalConjectures/Millenium/PvsNP/#ComplexityTheory___P_subset_NP","anchor":"ComplexityTheory___P_subset_NP","docHtml":"\n The theorem that P is a subset of NP.
\n\n This can be proven by observing that for any language in P,\nwe can construct a verifier that ignores the witness and simply runs the poly-time decider for the\nlanguage.
"},"ComplexityTheory.P_subset_coNP":{"url":"/FormalConjectures/Millenium/PvsNP/#ComplexityTheory___P_subset_coNP","anchor":"ComplexityTheory___P_subset_coNP","docHtml":"\n The theorem that P is a subset of coNP.
"},"NavierStokes.divergence":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___divergence","anchor":"NavierStokes___divergence","docHtml":"\n The divergence $\\nabla \\cdot v$ of a vector field $v : \\mathbb{R}^n \\to \\mathbb{R}^n$\nat a point $x$, computed as the trace of the Jacobian matrix.
\n\n In coordinates, $\\nabla \\cdot v = \\sum_i \\partial v_i / \\partial x_i$.
\n\n This is available as the notation ∇⬝ v. If v is not differentiable at x, then\nfderiv is the zero map, so this definition has the corresponding junk value $0$.
\n The divergence of a vector field is $0$ at points where fderiv has its junk value.
\n The divergence of the zero vector field is zero.
"},"NavierStokes.divergence_const":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___divergence_const","anchor":"NavierStokes___divergence_const","docHtml":"\n The divergence of a constant vector field is zero.
"},"NavierStokes.divergence_add":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___divergence_add","anchor":"NavierStokes___divergence_add","docHtml":"\n Divergence is additive at points where both vector fields are differentiable.
"},"NavierStokes.divergence_smul":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___divergence_smul","anchor":"NavierStokes___divergence_smul","docHtml":"\n Divergence commutes with scalar multiplication at differentiability points.
"},"NavierStokes.IsOnePeriodic":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___IsOnePeriodic","anchor":"NavierStokes___IsOnePeriodic","docHtml":"\n A function $f : \\mathbb{R}^n \\to \\alpha$ is 1-periodic if it is periodic in each\ncoordinate with period $1$, i.e. $f(x + e_i) = f(x)$ for each unit vector $e_i$.\nThis captures functions on the $n$-torus $\\mathbb{R}^n/\\mathbb{Z}^n$.
"},"NavierStokes.InitialVelocityCondition":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___InitialVelocityCondition","anchor":"NavierStokes___InitialVelocityCondition","docHtml":"\n Basic conditions on initial velocity field for the Navier-Stokes equations\nin $n$-dimensional space.
\n\n The initial velocity must be:
\n\n Divergence-free (incompressibility condition: $\\nabla \\cdot u_0 = 0$)
\n\n Smooth ($C^\\infty$)
\n\n These conditions apply regardless of spatial dimension.
"},"NavierStokes.InitialVelocityCondition.div_free":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___InitialVelocityCondition___div_free","anchor":"NavierStokes___InitialVelocityCondition___div_free","docHtml":"\n The initial velocity field is divergence-free (equation 2).\nThis is the incompressibility constraint for the fluid.
"},"NavierStokes.InitialVelocityCondition.smooth":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___InitialVelocityCondition___smooth","anchor":"NavierStokes___InitialVelocityCondition___smooth","docHtml":"\n The initial velocity field is smooth ($C^\\infty$ in all variables).
"},"NavierStokes.InitialVelocityConditionDecay":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___InitialVelocityConditionDecay","anchor":"NavierStokes___InitialVelocityConditionDecay","docHtml":"\n Initial velocity conditions for the Navier-Stokes problem on all of $\\mathbb{R}^n$.
\n\n In addition to being smooth and divergence-free, the velocity must decay\nfaster than any polynomial at spatial infinity (condition 4 in Fefferman's paper).
\n\n This condition ensures the velocity field has finite energy and reasonable\nbehavior as $\\lVert x \\rVert \\to \\infty$.
"},"NavierStokes.InitialVelocityConditionDecay.decay":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___InitialVelocityConditionDecay___decay","anchor":"NavierStokes___InitialVelocityConditionDecay___decay","docHtml":"\n All derivatives of u₀ decay faster than any polynomial (condition 4).\nFor any derivative order $m$ and any decay rate $K$, there exists a constant $C$\nsuch that $\\lVert \\partial^m u_0(x) \\rVert \\le C/(1+\\lVert x \\rVert)^K$.
"},"NavierStokes.InitialVelocityConditionPeriodic":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___InitialVelocityConditionPeriodic","anchor":"NavierStokes___InitialVelocityConditionPeriodic","docHtml":"\n Initial velocity conditions for the periodic Navier-Stokes problem on\n$\\mathbb{R}^n/\\mathbb{Z}^n$.
\n\n The velocity must be smooth, divergence-free, and 1-periodic in each coordinate\n(condition 8, part 1 in Fefferman's paper).
"},"NavierStokes.InitialVelocityConditionPeriodic.isOnePeriodic":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___InitialVelocityConditionPeriodic___isOnePeriodic","anchor":"NavierStokes___InitialVelocityConditionPeriodic___isOnePeriodic","docHtml":"\n The initial velocity is 1-periodic in each direction (condition 8, part 1).
"},"NavierStokes.ForceCondition":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___ForceCondition","anchor":"NavierStokes___ForceCondition","docHtml":"\n The basic smoothness condition on the external forcing term.
\n\n The force $f(x,t)$ must be smooth ($C^\\infty$) in both space and time variables\nfor $t \\ge 0$.
"},"NavierStokes.ForceCondition.smooth":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___ForceCondition___smooth","anchor":"NavierStokes___ForceCondition___smooth","docHtml":"\n The force is smooth on $\\mathbb{R}^n \\times [0,\\infty)$.
"},"NavierStokes.ForceConditionDecay":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___ForceConditionDecay","anchor":"NavierStokes___ForceConditionDecay","docHtml":"\n Force conditions for the Navier-Stokes problem on all of $\\mathbb{R}^n$.
\n\n The force must be smooth and decay faster than any polynomial\nin both space and time (condition 5 in Fefferman's paper).
"},"NavierStokes.ForceConditionDecay.decay":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___ForceConditionDecay___decay","anchor":"NavierStokes___ForceConditionDecay___decay","docHtml":"\n All derivatives of f decay faster than any polynomial in space and time (condition 5).\nFor any derivative order $m$ and any decay rate $K$, there exists $C$ such that\n$\\lVert \\partial^m_{x,t} f(x,t) \\rVert \\le C/(1+\\lVert x \\rVert+t)^K$ for\n$t \\ge 0$.
"},"NavierStokes.ForceConditionPeriodic":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___ForceConditionPeriodic","anchor":"NavierStokes___ForceConditionPeriodic","docHtml":"\n Force conditions for the periodic Navier-Stokes problem on\n$\\mathbb{R}^n/\\mathbb{Z}^n$.
\n\n The force must be smooth, 1-periodic in space, and decay in time\n(conditions 8, part 1 and 9 in Fefferman's paper).
"},"NavierStokes.ForceConditionPeriodic.isOnePeriodic":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___ForceConditionPeriodic___isOnePeriodic","anchor":"NavierStokes___ForceConditionPeriodic___isOnePeriodic","docHtml":"\n The force is 1-periodic in space for all times $t \\ge 0$ (condition 8, part 1).
"},"NavierStokes.ForceConditionPeriodic.decay":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___ForceConditionPeriodic___decay","anchor":"NavierStokes___ForceConditionPeriodic___decay","docHtml":"\n All derivatives of f decay faster than any polynomial in time (condition 9).
"},"NavierStokes.NavierStokesExistenceAndSmoothness":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___NavierStokesExistenceAndSmoothness","anchor":"NavierStokes___NavierStokesExistenceAndSmoothness","docHtml":"\n A solution (v, p) to the Navier-Stokes equations in n-dimensional space\nwith viscosity $\\nu$, initial velocity $u_0$, and external force $f$.
\n\n This structure captures the core requirements for a solution:
\n\n The velocity and pressure satisfy the Navier-Stokes PDE (equation 1)
\n\n The velocity remains divergence-free for all time (equation 2)
\n\n The initial condition is satisfied (equation 3)
\n\n The solution is smooth ($C^\\infty$) for all time $t \\ge 0$ (equations 6, 11)
\n\n The Navier-Stokes equation (equation 1):\n$\\partial v/\\partial t + (v \\cdot \\nabla)v = \\nu\\Delta v - \\nabla p + f$.
"},"NavierStokes.NavierStokesExistenceAndSmoothness.div_free":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___NavierStokesExistenceAndSmoothness___div_free","anchor":"NavierStokes___NavierStokesExistenceAndSmoothness___div_free","docHtml":"\n Incompressibility constraint (equation 2): $\\nabla \\cdot v = 0$ for all\n$x$ and $t \\ge 0$.
"},"NavierStokes.NavierStokesExistenceAndSmoothness.initial_condition":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___NavierStokesExistenceAndSmoothness___initial_condition","anchor":"NavierStokes___NavierStokesExistenceAndSmoothness___initial_condition","docHtml":"\n Initial condition (equation 3): $v(x,0) = u_0(x)$ for all $x$.
"},"NavierStokes.NavierStokesExistenceAndSmoothness.velocity_smooth":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___NavierStokesExistenceAndSmoothness___velocity_smooth","anchor":"NavierStokes___NavierStokesExistenceAndSmoothness___velocity_smooth","docHtml":"\n The velocity field is smooth ($C^\\infty$) on $\\mathbb{R}^n \\times [0,\\infty)$\n(conditions 6, 11).
"},"NavierStokes.NavierStokesExistenceAndSmoothness.pressure_smooth":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___NavierStokesExistenceAndSmoothness___pressure_smooth","anchor":"NavierStokes___NavierStokesExistenceAndSmoothness___pressure_smooth","docHtml":"\n The pressure field is smooth ($C^\\infty$) on $\\mathbb{R}^n \\times [0,\\infty)$\n(conditions 6, 11).
"},"NavierStokes.NavierStokesExistenceAndSmoothnessRn":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___NavierStokesExistenceAndSmoothnessRn","anchor":"NavierStokes___NavierStokesExistenceAndSmoothnessRn","docHtml":"\n A solution to the Navier-Stokes equations on all of $\\mathbb{R}^n$ with appropriate\ndecay and energy bounds.
\n\n In addition to the basic solution properties, we require:
\n\n The velocity is in $L^2$ at each time $t \\ge 0$ (finite kinetic energy)
\n\n The total energy remains bounded for all time (condition 7)
\n\n The velocity is square-integrable at each time $t \\ge 0$ (condition 7).
"},"NavierStokes.NavierStokesExistenceAndSmoothnessRn.globally_bounded_energy":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___NavierStokesExistenceAndSmoothnessRn___globally_bounded_energy","anchor":"NavierStokes___NavierStokesExistenceAndSmoothnessRn___globally_bounded_energy","docHtml":"\n The kinetic energy $\\int \\lVert v(x,t) \\rVert^2,dx$ remains uniformly bounded\nfor all time (condition 7), where the integral is the Lebesgue integral.
"},"NavierStokes.NavierStokesExistenceAndSmoothnessPeriodic":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___NavierStokesExistenceAndSmoothnessPeriodic","anchor":"NavierStokes___NavierStokesExistenceAndSmoothnessPeriodic","docHtml":"\n A solution to the Navier-Stokes equations on the $n$-torus $\\mathbb{R}^n/\\mathbb{Z}^n$.
\n\n The velocity must be 1-periodic in each spatial direction for all times (condition 10).\nThe pressure is also required to be 1-periodic, following the errata appended to the\nClay problem statement.
"},"NavierStokes.NavierStokesExistenceAndSmoothnessPeriodic.isOnePeriodic_velocity":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___NavierStokesExistenceAndSmoothnessPeriodic___isOnePeriodic_velocity","anchor":"NavierStokes___NavierStokesExistenceAndSmoothnessPeriodic___isOnePeriodic_velocity","docHtml":"\n The velocity is 1-periodic in space for all times $t \\ge 0$ (condition 10).
"},"NavierStokes.NavierStokesExistenceAndSmoothnessPeriodic.isOnePeriodic_pressure":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___NavierStokesExistenceAndSmoothnessPeriodic___isOnePeriodic_pressure","anchor":"NavierStokes___NavierStokesExistenceAndSmoothnessPeriodic___isOnePeriodic_pressure","docHtml":"\n The pressure is 1-periodic in space for all times $t \\ge 0$ (Clay errata).
"},"NavierStokes.navier_stokes_existence_and_smoothness_R3":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___navier_stokes_existence_and_smoothness_R3","anchor":"NavierStokes___navier_stokes_existence_and_smoothness_R3","docHtml":"\n (A) Existence and smoothness of Navier–Stokes solutions on ℝ³.
"},"NavierStokes.navier_stokes_existence_and_smoothness_periodic":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___navier_stokes_existence_and_smoothness_periodic","anchor":"NavierStokes___navier_stokes_existence_and_smoothness_periodic","docHtml":"\n (B) Existence and smoothness of Navier–Stokes solutions in ℝ³/ℤ³.
"},"NavierStokes.navier_stokes_breakdown_R3":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___navier_stokes_breakdown_R3","anchor":"NavierStokes___navier_stokes_breakdown_R3","docHtml":"\n (C) Breakdown of Navier–Stokes solutions on ℝ³.
"},"NavierStokes.navier_stokes_breakdown_periodic":{"url":"/FormalConjectures/Millenium/NavierStokes/#NavierStokes___navier_stokes_breakdown_periodic","anchor":"NavierStokes___navier_stokes_breakdown_periodic","docHtml":"\n (D) Breakdown of Navier–Stokes Solutions on ℝ³/ℤ³.
"},"RiemannHypothesis.riemannHypothesis":{"url":"/FormalConjectures/Millenium/RiemannHypothesis/#RiemannHypothesis___riemannHypothesis","anchor":"RiemannHypothesis___riemannHypothesis","docHtml":"\n The Riemann Hypothesis: all non-trivial zeros of the Riemann zeta function have real\npart $\\frac{1}{2}$. That is, if $\\zeta(s) = 0$, $s \\neq 1$, and $s$ is not a trivial zero\n$-2(n+1)$ for some $n \\in \\mathbb{N}$, then $\\operatorname{Re}(s) = \\frac{1}{2}$.
\n\n This is the official Millennium Prize Problem as posed by the\nClay Mathematics Institute.
\n\n This uses the RiemannHypothesis type from Mathlib, which is defined as\n∀ (s : ℂ), riemannZeta s = 0 → (¬∃ n : ℕ, s = -2 * (n + 1)) → s ≠ 1 → s.re = 1 / 2.
\n Let $\\chi$ be a Dirichlet character, trivialZeros is the set of trivial zeros of the\nDirichlet $L$-function of $\\chi$ which is always a set of non-positive integers.
\n $\\chi = 1$ then the Dirichlet $L$-function is the Riemann zeta function, having trivial\nzeroes at all negative even integers (exclude $0$).
\n\n $\\chi$ is odd, then the trivial zeroes are the negative odd integers.
\n\n $\\chi \\neq 1$ is even, then the trivial zeroes are the non-positive even integers.
\n\n The Generalized Riemann Hypothesis asserts that all the non-trivial zeros of the\nDirichlet $L$-function $L(\\chi, s)$ of a primitive Dirichlet character $\\chi$ have real part\n$\\frac{1}{2}$.
"},"GRH.implies_riemannHypothesis":{"url":"/FormalConjectures/Millenium/RiemannHypothesis/#GRH___implies_riemannHypothesis","anchor":"GRH___implies_riemannHypothesis","docHtml":"\n GRH for $\\chi = 1$ is RiemannHypothesis.
\n The predicate that the generalized Poincaré conjecture holds in dimension $n$, i.e. that\nany $n$-dimensional manifold that is homotopy equivalent to the sphere is in fact homeomorphic\nto the sphere.
"},"PoincareConjecture.poincare_conjecture":{"url":"/FormalConjectures/Millenium/Poincare/#PoincareConjecture___poincare_conjecture","anchor":"PoincareConjecture___poincare_conjecture","docHtml":"\n The Millenium Problem, solved by Grigori Perelman in 2003: the Poincaré Conjecture holds.
"},"PoincareConjecture.poincare_conjecture.variants.dimension_two":{"url":"/FormalConjectures/Millenium/Poincare/#PoincareConjecture___poincare_conjecture___variants___dimension_two","anchor":"PoincareConjecture___poincare_conjecture___variants___dimension_two","docHtml":"\n The Generalized Poincaré Conjecture holds for surfaces.
"},"PoincareConjecture.poincare_conjecture.variants.dimension_ge_five":{"url":"/FormalConjectures/Millenium/Poincare/#PoincareConjecture___poincare_conjecture___variants___dimension_ge_five","anchor":"PoincareConjecture___poincare_conjecture___variants___dimension_ge_five","docHtml":"\n The Generalized Poincaré Conjecture holds for dimensions at least 5.
"},"PoincareConjecture.poincare_conjecture.variants.dimension_four":{"url":"/FormalConjectures/Millenium/Poincare/#PoincareConjecture___poincare_conjecture___variants___dimension_four","anchor":"PoincareConjecture___poincare_conjecture___variants___dimension_four","docHtml":"\n The Generalized Poincaré Conjecture holds in dimension 4.
"},"PoincareConjecture.SmoothConjectureFor":{"url":"/FormalConjectures/Millenium/Poincare/#PoincareConjecture___SmoothConjectureFor","anchor":"PoincareConjecture___SmoothConjectureFor","docHtml":"\n The predicate that the smooth Poincaré conjecture holds in dimension $n$.
"},"PoincareConjecture.poincare_conjecture.variants.smooth_for_three":{"url":"/FormalConjectures/Millenium/Poincare/#PoincareConjecture___poincare_conjecture___variants___smooth_for_three","anchor":"PoincareConjecture___poincare_conjecture___variants___smooth_for_three","docHtml":"\n A reformulation of the Millenium Problem in terms of smooth 3-folds.
"},"PoincareConjecture.poincare_conjecture.variants.smooth_implication":{"url":"/FormalConjectures/Millenium/Poincare/#PoincareConjecture___poincare_conjecture___variants___smooth_implication","anchor":"PoincareConjecture___poincare_conjecture___variants___smooth_implication","docHtml":"\n The smooth formulation of the Millenium Problem implies the general case. This follows from\nthe fact that every topological 3-fold admits a smooth structure [mo296171].
"},"PoincareConjecture.SmoothTrueValues":{"url":"/FormalConjectures/Millenium/Poincare/#PoincareConjecture___SmoothTrueValues","anchor":"PoincareConjecture___SmoothTrueValues","docHtml":"\n The values at which the smooth version of the conjecture is known to hold.
"},"PoincareConjecture.poincare_conjecture.variants.smooth_known_cases":{"url":"/FormalConjectures/Millenium/Poincare/#PoincareConjecture___poincare_conjecture___variants___smooth_known_cases","anchor":"PoincareConjecture___poincare_conjecture___variants___smooth_known_cases","docHtml":"\n The smooth version of the Poincaré conjecture is known to hold in dimensions\n$1, 2, 3, 5, 6, 12, 56, 61$. See [Wang2017].
"},"PoincareConjecture.poincare_conjecture.variants.smooth_dimension_four":{"url":"/FormalConjectures/Millenium/Poincare/#PoincareConjecture___poincare_conjecture___variants___smooth_dimension_four","anchor":"PoincareConjecture___poincare_conjecture___variants___smooth_dimension_four","docHtml":"\n The four dimensional case of the smooth version of the conjecture is still open.\nSee [Wang2017].
"},"PoincareConjecture.poincare_conjecture.variants.smooth_other_cases":{"url":"/FormalConjectures/Millenium/Poincare/#PoincareConjecture___poincare_conjecture___variants___smooth_other_cases","anchor":"PoincareConjecture___poincare_conjecture___variants___smooth_other_cases","docHtml":"\n It is conjectured that the only values of $n > 4$ for which the smooth version of the\nconjecture holds are $n = 5, 6, 12, 56, 61$. See Conjecture 1.17 in [Wang2017].
"},"Mathoverflow75792.Reachable":{"url":"/FormalConjectures/Mathoverflow/«75792»/#Mathoverflow75792___Reachable","anchor":"Mathoverflow75792___Reachable","docHtml":"\n The inductively defined predicate that m is reachable in n steps.
\n The inductively defined predicate that m is reachable in n steps.
\n The inductively defined predicate that m is reachable in n steps.
\n The inductively defined predicate that m is reachable in n steps.
\n The (Mahler-Popken) complexity of n:\nthe minimum number of 1s needed to express a given number using only addition and\nmultiplication. E.g. 2 = 1 + 1, so complexity 2 = 2.
\n5^6 = 15625 = 1 + 2^3 * 3^2 * (1 + 2^3 * 3^3)!
\n Is 5n the complexity of 5^n for 0 < n? Answer: No.
\n Is 3n the complexity of 3^n for 0 < n? Answer: Yes, by John Selfridge.
\n Reference: https://arxiv.org/abs/1207.4841
"},"Mathoverflow75792.complexity_two_pow":{"url":"/FormalConjectures/Mathoverflow/«75792»/#Mathoverflow75792___complexity_two_pow","anchor":"Mathoverflow75792___complexity_two_pow","docHtml":"\n Is 2n the complexity of 2^n for 0 < n?
\n The predicate that all coefficients of a polynomial are either zero or one.\nP.coeffs is the finite set of all P.\nSo IsZeroOne P means that every nonzero coefficient of P is equal to 1.\nNote that zero coefficients are not included in P.coeffs.
\n Let $P(x), Q(x) ∈ ℝ[x]$ be two monic polynomials with non-negative coefficients.\nIf $R(x) = P(x)Q(x)$ is a $0,1$ polynomial (coefficients only from ${0,1}$), then $P(x)$ and $Q(x)$\nare also $0, 1$ polynomials.
"},"Mathoverflow339137.mathoverflow_339137_probabilistic":{"url":"/FormalConjectures/Mathoverflow/«339137»/#Mathoverflow339137___mathoverflow_339137_probabilistic","anchor":"Mathoverflow339137___mathoverflow_339137_probabilistic","docHtml":"\n Green's Open Problem 28 is the probabilistic reformulation of Mathoverflow 339137.
\n\n Suppose that $X, Y$ are two finitely-supported independent random variables taking integer values,\nand such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly\ndistributed on their ranges?
\n\n Mathematically, this equivalence is established via Probability Generating Functions (PGFs),\nshifting the support to $\\mathbb{N}$, and appropriately scaling the coefficients.
"},"Mathoverflow1973.unitSphere":{"url":"/FormalConjectures/Mathoverflow/«1973»/#Mathoverflow1973___unitSphere","anchor":"Mathoverflow1973___unitSphere","docHtml":"\n The unit n-sphere, defined as Metric.sphere 0 1 in EuclideanSpace ℝ (Fin (n + 1)).
\n Does the 6-sphere admit a complex structure, i.e. an atlas of holomorphically compatible charts\nrelating it to EuclideanSpace ℂ (Fin 3)?
\n If $2^x$ and $3^x$ are integers, then $x$ must be an integer.
"},"Mathoverflow17560.mathoverflow_17560.variants.all_nats":{"url":"/FormalConjectures/Mathoverflow/«17560»/#Mathoverflow17560___mathoverflow_17560___variants___all_nats","anchor":"Mathoverflow17560___mathoverflow_17560___variants___all_nats","docHtml":"\n If for each natural number $n$ the number $n^x$ is an integer then $x$ must also be an integer.
"},"Mathoverflow17560.mathoverflow_17560.variants.with_5":{"url":"/FormalConjectures/Mathoverflow/«17560»/#Mathoverflow17560___mathoverflow_17560___variants___with_5","anchor":"Mathoverflow17560___mathoverflow_17560___variants___with_5","docHtml":"\n If $2^x$, $3^x$ and $5^x$ are integers, then $x$ must be an integer.
"},"Mathoverflow21003.mathoverflow_21003":{"url":"/FormalConjectures/Mathoverflow/«21003»/#Mathoverflow21003___mathoverflow_21003","anchor":"Mathoverflow21003___mathoverflow_21003","docHtml":"\n Is there any polynomial $f(x, y) \\in \\mathbb{Q}[x, y]$ such that\n$f : \\mathbb{Q} \\times \\mathbb{Q} \\rightarrow \\mathbb{Q}$ is a bijection?
"},"Mathoverflow486451.exists_semiring_unique_left_maximal_not_unique_right_maximal":{"url":"/FormalConjectures/Mathoverflow/«486451»/#Mathoverflow486451___exists_semiring_unique_left_maximal_not_unique_right_maximal","anchor":"Mathoverflow486451___exists_semiring_unique_left_maximal_not_unique_right_maximal","docHtml":"\n There exists a semiring with a unique left maximal ideal but more than one right maximal ideals.
"},"Mathoverflow486451.exists_semiring_unique_left_right_maximal_ne":{"url":"/FormalConjectures/Mathoverflow/«486451»/#Mathoverflow486451___exists_semiring_unique_left_right_maximal_ne","anchor":"Mathoverflow486451___exists_semiring_unique_left_right_maximal_ne","docHtml":"\n There exists a semiring with a unique left maximal ideal and a unique right maximal ideal\nwhich are not the same as sets.
\n\n This has been shown by Goran Žužić and Moritz Firsching using an experimental pipeline:\nAn example is the monoid algebra of the monoid of maps from $\\mathbb{N}$ to $\\mathbb{N}$\nover $\\mathbb{N}$.
"},"Mathoverflow235893.IsConnectedMap":{"url":"/FormalConjectures/Mathoverflow/«235893»/#Mathoverflow235893___IsConnectedMap","anchor":"Mathoverflow235893___IsConnectedMap","docHtml":"\n For topological spaces $X$ and $Y$ we say a function $f : X → Y$ is
\n By a standard result, every continuous map is connected
"},"Mathoverflow235893.isConnected_iff_ordConnected_and_nonempty":{"url":"/FormalConjectures/Mathoverflow/«235893»/#Mathoverflow235893___isConnected_iff_ordConnected_and_nonempty","anchor":"Mathoverflow235893___isConnected_iff_ordConnected_and_nonempty","docHtml":"\n A set in $\\mathbb{R}$ is connected if and only if it is order-connected and non-empty.
"},"Mathoverflow235893.isConnectedMap_symm_of_R":{"url":"/FormalConjectures/Mathoverflow/«235893»/#Mathoverflow235893___isConnectedMap_symm_of_R","anchor":"Mathoverflow235893___isConnectedMap_symm_of_R","docHtml":"\n If $f : \\mathbb{R} \\to \\mathbb{R}$ is a connected bijection, then its inverse is also a connected bijection.
"},"Mathoverflow235893.isConnectedMap_comp":{"url":"/FormalConjectures/Mathoverflow/«235893»/#Mathoverflow235893___isConnectedMap_comp","anchor":"Mathoverflow235893___isConnectedMap_comp","docHtml":"\n The composition of two connected maps is a connected map.
"},"Mathoverflow235893.isConnectedMap_homeomorph":{"url":"/FormalConjectures/Mathoverflow/«235893»/#Mathoverflow235893___isConnectedMap_homeomorph","anchor":"Mathoverflow235893___isConnectedMap_homeomorph","docHtml":"\n A homeomorphism is a connected map.
"},"Mathoverflow235893.isConnectedMap_symm_of_E1":{"url":"/FormalConjectures/Mathoverflow/«235893»/#Mathoverflow235893___isConnectedMap_symm_of_E1","anchor":"Mathoverflow235893___isConnectedMap_symm_of_E1","docHtml":"\n If $f : \\mathbb{R}^1 \\to \\mathbb{R}^1$ is a connected bijection, then its inverse is also a connected bijection.
"},"Mathoverflow235893.mathoverflow_235893":{"url":"/FormalConjectures/Mathoverflow/«235893»/#Mathoverflow235893___mathoverflow_235893","anchor":"Mathoverflow235893___mathoverflow_235893","docHtml":"\n Assume for $n>1$, $f:\\mathbb{R}^n\\to\\mathbb{R}^n$ is a bijection, where $\\mathbb{R}^n$ is equipped\nwith the standard topology. Does the connectedness of (the induced power set map) $f$ imply\nthat of $f^{-1}$?
"},"Mathoverflow235893.mathoverflow_260589":{"url":"/FormalConjectures/Mathoverflow/«235893»/#Mathoverflow235893___mathoverflow_260589","anchor":"Mathoverflow235893___mathoverflow_260589","docHtml":"\n There exists a connected bijection ℝ → ℝ^2 where the inverse is not connected,\nproven in mathoverflow/260589 by user\nGro-Tsen.
"},"Mathoverflow347178.mathoverflow_347178":{"url":"/FormalConjectures/Mathoverflow/«347178»/#Mathoverflow347178___mathoverflow_347178","anchor":"Mathoverflow347178___mathoverflow_347178","docHtml":"\n Let $f : \\mathbb R^n \\to \\mathbb R, n \\geq 2$ be a $C^1$ function. Is it true that\n$$\\sup_{x \\in \\mathbb R^n}f(x) = \\sup_{x\\in \\mathbb R^n} f(x+\\nabla f(x))$$?
"},"Mathoverflow347178.mathoverflow_347178.variants.bounded_iff":{"url":"/FormalConjectures/Mathoverflow/«347178»/#Mathoverflow347178___mathoverflow_347178___variants___bounded_iff","anchor":"Mathoverflow347178___mathoverflow_347178___variants___bounded_iff","docHtml":"\n Let $f : \\mathbb R^n \\to \\mathbb R, n \\geq 2$ be a $C^1$ function. Is the boundedness of\n$\\sup_{x \\in \\mathbb R^n}f(x)$ and $\\sup_{x\\in \\mathbb R^n} f(x+\\nabla f(x))$ equivalent?
"},"Mathoverflow347178.mathoverflow_347178.variants.bounded_only":{"url":"/FormalConjectures/Mathoverflow/«347178»/#Mathoverflow347178___mathoverflow_347178___variants___bounded_only","anchor":"Mathoverflow347178___mathoverflow_347178___variants___bounded_only","docHtml":"\n Let $f : \\mathbb R^n \\to \\mathbb R, n \\geq 2$ be a $C^1$ function. Does the equality\n$$\\sup_{x \\in \\mathbb R^n}f(x) = \\sup_{x\\in \\mathbb R^n} f(x+\\nabla f(x))$$\nhold when both suprema are finite?
"},"Mathoverflow34145.Rectangle":{"url":"/FormalConjectures/Mathoverflow/«34145»/#Mathoverflow34145___Rectangle","anchor":"Mathoverflow34145___Rectangle","docHtml":"\n A rectangle is specified by its width, height, starting point, and rotation.\nThe rectangle is assumed to start in the lower left corner. For example, the unit square\n${ (x, y) \\mid 0 \\le x \\le 1, 0 \\le y \\le 1 }$ is specified as ⟨1, 1, (0, 0), 0⟩
\n A rectangle is specified by its width, height, starting point, and rotation.\nThe rectangle is assumed to start in the lower left corner. For example, the unit square\n${ (x, y) \\mid 0 \\le x \\le 1, 0 \\le y \\le 1 }$ is specified as ⟨1, 1, (0, 0), 0⟩
\n A rectangle is specified by its width, height, starting point, and rotation.\nThe rectangle is assumed to start in the lower left corner. For example, the unit square\n${ (x, y) \\mid 0 \\le x \\le 1, 0 \\le y \\le 1 }$ is specified as ⟨1, 1, (0, 0), 0⟩
\n A rectangle is specified by its width, height, starting point, and rotation.\nThe rectangle is assumed to start in the lower left corner. For example, the unit square\n${ (x, y) \\mid 0 \\le x \\le 1, 0 \\le y \\le 1 }$ is specified as ⟨1, 1, (0, 0), 0⟩
\n A rectangle is specified by its width, height, starting point, and rotation.\nThe rectangle is assumed to start in the lower left corner. For example, the unit square\n${ (x, y) \\mid 0 \\le x \\le 1, 0 \\le y \\le 1 }$ is specified as ⟨1, 1, (0, 0), 0⟩
\n A combination of a rotation and a translation to map the standard rectangle to the desired\nrectangle.
"},"Mathoverflow34145.rigidMotion_test":{"url":"/FormalConjectures/Mathoverflow/«34145»/#Mathoverflow34145___rigidMotion_test","anchor":"Mathoverflow34145___rigidMotion_test"},"Mathoverflow34145.scale":{"url":"/FormalConjectures/Mathoverflow/«34145»/#Mathoverflow34145___scale","anchor":"Mathoverflow34145___scale","docHtml":"\n A scaling to map the unit square to a standard rectangle.
"},"Mathoverflow34145.unitSquare":{"url":"/FormalConjectures/Mathoverflow/«34145»/#Mathoverflow34145___unitSquare","anchor":"Mathoverflow34145___unitSquare","docHtml":"\n The unit square.
"},"Mathoverflow34145.Rectangle.toSet":{"url":"/FormalConjectures/Mathoverflow/«34145»/#Mathoverflow34145___Rectangle___toSet","anchor":"Mathoverflow34145___Rectangle___toSet","docHtml":"\n Converts a rectangle to a set in ℝ × ℝ.
\n The standard Lebesgue measure on ℝ².
\nlbMeasure is invariant under rigidMotion start θ.
\nlbMeasure is scaled by scale.
\n The Lebesgue measure of the unit square is 1.
\n The Lebesgue measure of the a rectangle r is r.width * r.height
\n The areas of the required rectangles sum to 1.
"},"Mathoverflow34145.Configuration":{"url":"/FormalConjectures/Mathoverflow/«34145»/#Mathoverflow34145___Configuration","anchor":"Mathoverflow34145___Configuration","docHtml":"\n A configuration of rectangles of sides 1 / (n + 1) and 1 / (n + 2).
\n A configuration of rectangles of sides 1 / (n + 1) and 1 / (n + 2).
\n A configuration of rectangles of sides 1 / (n + 1) and 1 / (n + 2).
\n A configuration of rectangles of sides 1 / (n + 1) and 1 / (n + 2).
\n A \"packing\" means that the interiors of any two rectangles are disjoint.
"},"Mathoverflow34145.rectangles_cover_unit_square":{"url":"/FormalConjectures/Mathoverflow/«34145»/#Mathoverflow34145___rectangles_cover_unit_square","anchor":"Mathoverflow34145___rectangles_cover_unit_square","docHtml":"\n Can a unit square be covered by rectangles of width 1 / (n + 1) and height 1 / (n + 2)?
\n Equivalently, can a unit square be packed with rectangles of width 1 / (n + 1) and height\n1 / (n + 2)?
\n It is known that packing the rectangles into a square of side length 133/132 is possible.
\n Reference: https://www.sciencedirect.com/science/article/pii/0097316594901163
"},"Mathoverflow34145.rectangles_pack_square_501_div_500":{"url":"/FormalConjectures/Mathoverflow/«34145»/#Mathoverflow34145___rectangles_pack_square_501_div_500","anchor":"Mathoverflow34145___rectangles_pack_square_501_div_500","docHtml":"\n It is known that packing the rectangles into a square of side length 501/500 is possible.
\n Reference: https://www.sciencedirect.com/science/article/pii/S0167506008706009
"},"Mathoverflow31809.mathoverflow_31809":{"url":"/FormalConjectures/Mathoverflow/«31809»/#Mathoverflow31809___mathoverflow_31809","anchor":"Mathoverflow31809___mathoverflow_31809","docHtml":"\n Does there exist a category that is pretriangulated but not triangulated?
"},"Mathoverflow10799.μ":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799______","anchor":"Mathoverflow10799______","docHtml":"\n Start with a set $X={1,2,...,n}$ of $n$ elements and the family $2^X$ of all subsets of $X$.\nFor a real number $p$ between zero and one, we consider a probability distribution $\\mu_p$ on\n$2^X$ where the probability that $i \\in S$ is $p$, independently for different $i$'s.\nThus for $p=1/2$ we get the uniform probability distribution.
\n\n For $S \\subseteq {0, \\ldots, n-1}$, its probability is $p^{|S|} (1-p)^{n - |S|}$.
"},"Mathoverflow10799.μ_half_eq_uniform":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799_______half_eq_uniform","anchor":"Mathoverflow10799_______half_eq_uniform","docHtml":"\n For $p = 1/2$, the $p$-biased measure is the uniform distribution:\n$\\mu_{1/2}(S) = (1/2)^n$ for every $S \\subseteq [n]$.
"},"Mathoverflow10799.μFamily":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799______Family","anchor":"Mathoverflow10799______Family","docHtml":"\n The $p$-biased measure of a family $\\mathcal F \\subseteq 2^{[n]}$,\ni.e. $\\mu_p(\\mathcal F) = \\sum_{S \\in \\mathcal F} \\mu_p(S)$.
"},"Mathoverflow10799.boundaryCount":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799___boundaryCount","anchor":"Mathoverflow10799___boundaryCount","docHtml":"\n Given a family $F$, for a subset $S$ of $X$, we write $h(S)$ as the number of subsets $T$ in $X$\nsuch that\n(1) $T$ differs from $S$ in exactly one element\n(2) Exactly one set among $S$ and $T$ belongs to $F$.
"},"Mathoverflow10799.boundaryCount_equiv":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799___boundaryCount_equiv","anchor":"Mathoverflow10799___boundaryCount_equiv","docHtml":"\n Test lemma showing that boundaryCount is equivalent to counting subsets $T$\nthat differ from $S$ in exactly one element and exactly one of $S, T$ belongs to $F$.
\n The edge-boundary of $F$ is the expectation of $h(S)$ (according to $\\mu_p$) over all\nsubsets $S$ of $X$. It is denoted by $I^p(F)$.
"},"Mathoverflow10799.IsMonotoneIncreasing":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799___IsMonotoneIncreasing","anchor":"Mathoverflow10799___IsMonotoneIncreasing","docHtml":"\n A family $F$ of subsets of $2^X$ is monotone increasing if when $S$ belongs to $F$ and $T$\ncontains $S$ then $T$ also belongs to $F$. (Monotone increasing families also also called \"filtes\"\nand \"up-families\".) From now on we will restrict our attention to the case of monotone increasing\nfamilies.
\n\n This is Mathlib's IsUpperSet applied to the coercion of $\\mathcal F$ to a set,\nusing the fact that ≤ on Finset is ⊆.
\n We say that a family is optimal for $\\mu_p$ if the isoperimetric inequality (IR) is sharp up to a\nmultiplicative constant $1000 \\log (1/p)$.
\n\n Since the lower bound from (IR) is\n$$\\frac{\\mu_p(\\mathcal F) \\cdot \\log(1/\\mu_p(\\mathcal F))}{p \\cdot \\log(1/p)},$$\nmultiplying by $1000 \\log(1/p)$ gives the condition\n$$I^p(\\mathcal F) \\le \\frac{1000}{p} \\cdot \\mu_p(\\mathcal F) \\cdot \\log \\frac{1}{\\mu_p(\\mathcal F)}.$$
"},"Mathoverflow10799.mathoverflow_10799":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799___mathoverflow_10799","anchor":"Mathoverflow10799___mathoverflow_10799","docHtml":"\n Problem: For every monotone increasing family $F$, given an interval $[s,t]$ of real numbers so\nthat $t/s > 1000 \\log n$ we have some $p$ in the interval $[s,t]$ so that $F$ is optimal with\nrespect to $\\mu_p$.
\n\n This was a \"missing lemma\" in the work of Kahn and Kalai on threshold behavior of monotone\nproperties. The related Kahn–Kalai conjecture was\nsettled by Park and Pham.
\n\nThis conjecture is false without the additional assumption $\\mu_t(F) = 1/2$.\nA counterexample was found by Shlomo Perles (April 7, 2026).
"},"Mathoverflow10799.mathoverflow_10799.variants.kahn_kalai_conjecture_7":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799___mathoverflow_10799___variants___kahn_kalai_conjecture_7","anchor":"Mathoverflow10799___mathoverflow_10799___variants___kahn_kalai_conjecture_7","docHtml":"\n Conjecture 7 from Kahn–Kalai 2006: the same statement as the original\nconjecture, but with the additional assumption that $t$ is the critical probability for $F$,\nnamely $\\mu_t(F) = 1/2$.
"},"Mathoverflow10799.mathoverflow_10799.variants.weak_kahn_kalai":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799___mathoverflow_10799___variants___weak_kahn_kalai","anchor":"Mathoverflow10799___mathoverflow_10799___variants___weak_kahn_kalai","docHtml":"\n Weaker version proven by Kahn–Kalai: the same conclusion holds when $1000 \\log n$ is replaced by\n$C_\\varepsilon , n^\\varepsilon$ for every fixed $\\varepsilon > 0$.
"},"Mathoverflow10799.discrete_isoperimetric_inequality":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799___discrete_isoperimetric_inequality","anchor":"Mathoverflow10799___discrete_isoperimetric_inequality","docHtml":"\n Now a famous isoperimetric relation asserts that\n(IR) $I^p(F) \\ge \\frac{1}{p} \\mu_p(F) \\log_p \\mu_p(F)$\nThis relation is true for every family $F$ and every $p$. It is especially famous and simple when\n$p=1/2$ and $\\mu_p(F)=1/2$. In this case, it says that given a set of half the vertices of the\ndiscrete cube $2^X$, the number of edges between $F$ and its complement is at least $2^{n-1}$.
\n\nReal.logb p m to represent the logarithm base $p$ directly.\nThe factor of $p$ in the denominator (equivalent to $1/p$ in front) is consistent with the\ndefinition of IsOptimal used in the counterexample proof.
\n The $p$-biased measure is a probability distribution: it sums to $1$ over all subsets.\nThis is the binomial identity $(p + (1-p))^n = 1$.
"},"Mathoverflow10799.μFamily_univ":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799______Family_univ","anchor":"Mathoverflow10799______Family_univ","docHtml":"\n The measure of the full power set is $1$.
"},"Mathoverflow10799.boundaryCount_empty":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799___boundaryCount_empty","anchor":"Mathoverflow10799___boundaryCount_empty","docHtml":"\n The boundary count is zero for the empty family (no set is in $\\mathcal F$).
"},"Mathoverflow10799.edgeBoundary_empty":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799___edgeBoundary_empty","anchor":"Mathoverflow10799___edgeBoundary_empty","docHtml":"\n The edge boundary is zero for the empty family.
"},"Mathoverflow10799.boundaryCount_univ":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799___boundaryCount_univ","anchor":"Mathoverflow10799___boundaryCount_univ","docHtml":"\n The boundary count is zero for the full family (every set is in $\\mathcal F$).
"},"Mathoverflow10799.edgeBoundary_univ":{"url":"/FormalConjectures/Mathoverflow/«10799»/#Mathoverflow10799___edgeBoundary_univ","anchor":"Mathoverflow10799___edgeBoundary_univ","docHtml":"\n The edge boundary is zero for the full family.
"},"Equidistribution.IsAccumulationPoint":{"url":"/FormalConjectures/Books/UniformDistributionOfSequences/Equidistribution/#Equidistribution___IsAccumulationPoint","anchor":"Equidistribution___IsAccumulationPoint","docHtml":"\n A point x is an accumulation point of a sequence s_0, s_1, ...\nif any neighbourhood of x contains a point of the sequence distinct\nfrom x.
\n If a point x is an accumulation point of a sequence s_0, s_1, ... then\nthere is a subsequence of s that tends to x
\n The sequence (3/2)^n is equidistributed modulo 1.
\n For any transcendental number x, the sequence x * (3 / 2) ^ n is\nequidistributed modulo 1.
\n The sequence (3/2)^n has infinitely many accumulation points modulo 1.
\n Find an accumulation point of the sequence (3/2)^n modulo 1.
\n There is an accumulation point of the sequence (3/2)^n modulo 1.
\n Does the series\n$$\n\\sum_{n=1}^{\\infty} \\frac{\\left(\\frac{2}{3} + \\frac{1}{3}\\sin n\\right)^n}{n}\n$$\nconverge?
\n\n After computing approximately $10^7$ terms, the partial sums approximate $2.163$.
\n\n See https://arxiv.org/abs/2007.11017 for a proof of the convergence,\nrelying on an irrationality measure for pi.
\n\n Also see\nhttps://github.com/AxiomMath/gdm-formal-conjectures/blob/main/docs/BorweinSineSeries.md\nfor a partial formalization of the conjecture,\nconditional on such an irrationality measure of pi (cf https://arxiv.org/abs/1912.06345).
"},"Bugeaud.Spectrum":{"url":"/FormalConjectures/Books/BugeaudDistributionModuloOne/Problem10_4/#Bugeaud___Spectrum","anchor":"Bugeaud___Spectrum","docHtml":"\n The spectrum of a sequence $(x_n)
\n Problem 10.4. Let $\\xi$ be a non-zero real number and $\\alpha > 1$ be a real\nnumber. The spectrum of the sequence $(\\xi \\alpha^n)_{n \\ge 1}$ is at most\ncountable. Posed by Mendès France [Men73].
"},"Bugeaud.problem_10_7":{"url":"/FormalConjectures/Books/BugeaudDistributionModuloOne/Problem10_7/#Bugeaud___problem_10_7","anchor":"Bugeaud___problem_10_7","docHtml":"\n Problem 10.7. Let $\\varepsilon$ be a positive real number. Are there arbitrarily\nlarge real numbers $\\alpha$ such that $\\alpha$ is not a Pisot number and all the\nfractional parts ${\\alpha^n}$, $n \\ge 1$, are lying in an interval of length\n$\\varepsilon / \\alpha$? [Bug12b]
"},"Bugeaud.problem_10_1":{"url":"/FormalConjectures/Books/BugeaudDistributionModuloOne/IntDistanceDistribution/#Bugeaud___problem_10_1","anchor":"Bugeaud___problem_10_1","docHtml":"\n Problem 10.1. Are there a transcendental number $\\alpha$ and a positive real\nnumber $\\xi$ such that $\\lVert \\xi \\alpha^n \\rVert$ tends to~$0$ as~$n$ tends to infinity? [Har19]\n(Trivial for $|\\alpha| < 1$)
"},"Bugeaud.problem_10_2":{"url":"/FormalConjectures/Books/BugeaudDistributionModuloOne/IntDistanceDistribution/#Bugeaud___problem_10_2","anchor":"Bugeaud___problem_10_2","docHtml":"\n Problem 10.2. To prove that $\\lVert e^n \\rVert$ does not tend to 0 as n tends to\ninfinity.
"},"Bugeaud.problem_10_3":{"url":"/FormalConjectures/Books/BugeaudDistributionModuloOne/IntDistanceDistribution/#Bugeaud___problem_10_3","anchor":"Bugeaud___problem_10_3","docHtml":"\n Problem 10.3. To prove that there exists a positive real number~$c$ such\nthat $\\lVert e^n \\rVert > e^{−cn}$, for every~$n \\ge 1$. Posed by Mahler [Mah53].
"},"Bugeaud.waldschmidt":{"url":"/FormalConjectures/Books/BugeaudDistributionModuloOne/IntDistanceDistribution/#Bugeaud___waldschmidt","anchor":"Bugeaud___waldschmidt","docHtml":"\n Waldschmidt [Wal03] conjectured that a stronger result holds, namely\nthat there exists a positive real number~$c$ such that $\\lVert e^n \\rVert > n^{−c}$ for\nevery~$n \\ge 1$. This is supported by metrical results [Kok45].
"},"Bugeaud.problem_10_3_of_waldschmidt":{"url":"/FormalConjectures/Books/BugeaudDistributionModuloOne/IntDistanceDistribution/#Bugeaud___problem_10_3_of_waldschmidt","anchor":"Bugeaud___problem_10_3_of_waldschmidt","docHtml":"\n Waldschmidt's conjecture is stronger than Mahler's: since $\\log n \\le n$ for $n \\ge 1$,\nthe polynomial lower bound $n^{-c}$ dominates the exponential lower bound $e^{-cn}$.
"},"Bugeaud.problem_10_5":{"url":"/FormalConjectures/Books/BugeaudDistributionModuloOne/Problem10_5/#Bugeaud___problem_10_5","anchor":"Bugeaud___problem_10_5","docHtml":"\n Problem 10.5 (first part). Let $\\mathbb{K}$ be a real number field. Then, for any\n$\\varepsilon > 0$, there exists a lacunary sequence $(t_n)
\n Problem 10.5 (\"moreover\" clause). With the same hypotheses as problem_10_5, the\nsequence $(t_n)$ can be chosen so that, for any real $\\xi$ not in $\\mathbb{K}$, each\nsubinterval of $[0, 1]$ of length $\\varepsilon$ contains a limit point of the sequence\n$({\\xi t_n})_{n \\ge 1}$. This is strictly stronger than problem_10_5: the limsup\nbound is the special case at the subinterval $[1 - \\varepsilon, 1]$.
\n The \"moreover\" form of Problem 10.5 implies the first part: applying the cluster-point\ndensity to the subinterval $[1 - \\varepsilon, 1]$ yields the required lower bound on the\nlimsup.
"},"Erdos779.erdos_779":{"url":"/FormalConjectures/ErdosProblems/«779»/#Erdos779___erdos_779","anchor":"Erdos779___erdos_779","docHtml":"\n A Conjecture of Marian Deaconescu, see p.120 in https://doi.org/10.2307/2975810
\n\n [Needed to index shift in order to avoid trivial case $n = 0$,\nwhere the conjecture is trivially false.]
"},"Erdos90.unitDistanceCounts":{"url":"/FormalConjectures/ErdosProblems/«90»/#Erdos90___unitDistanceCounts","anchor":"Erdos90___unitDistanceCounts","docHtml":"\n The set of all possible numbers of unit distances for a configuration of $n$ points.
"},"Erdos90.unitDistanceCounts_BddAbove":{"url":"/FormalConjectures/ErdosProblems/«90»/#Erdos90___unitDistanceCounts_BddAbove","anchor":"Erdos90___unitDistanceCounts_BddAbove","docHtml":"\n This lemma confirms that the set of possible unit distance counts is bounded above, which\nensures that taking the supremum (sSup) is a well-defined operation. The trivial upper bound is\nthe total number of pairs of points, $\\binom{n}{2}$.
\n The maximum number of unit distances determined by any set of $n$ points in the plane.\nThis function is often denoted as $u(n)$ in combinatorics.
"},"Erdos90.erdos_90":{"url":"/FormalConjectures/ErdosProblems/«90»/#Erdos90___erdos_90","anchor":"Erdos90___erdos_90","docHtml":"\n Does every set of $n$ distinct points in $\\mathbb{R}^2$ contain at most\n$n^{1+O(\\frac{1}{\\log\\log n})}$ many pairs which are distance $1$ apart?
\n\n This was\ndisproved\nby an internal model at OpenAI, which constructed (for infinitely many $n$) a set $P$ of $n$ points\nin $\\mathbb{R}^2$ such that the number of unit distance pairs in $P$ is at least $n^{1+c}$, where\n$c > 0$ is an absolute constant.
"},"Erdos943.erdos_943":{"url":"/FormalConjectures/ErdosProblems/«943»/#Erdos943___erdos_943","anchor":"Erdos943___erdos_943","docHtml":"\n Let $A$ be the set of powerful numbers. Is is true that $1_A\\ast 1_A(n)=n^{o(1)}$ for every $n$?
"},"Erdos97.HasNEquidistantPointsAt":{"url":"/FormalConjectures/ErdosProblems/«97»/#Erdos97___HasNEquidistantPointsAt","anchor":"Erdos97___HasNEquidistantPointsAt","docHtml":"\n A set of points $A$ has n equidistant points at $p$\nif there exist at least $n$ other points in $A$ that are equidistant from $p$.
"},"Erdos97.HasNEquidistantPointsOn":{"url":"/FormalConjectures/ErdosProblems/«97»/#Erdos97___HasNEquidistantPointsOn","anchor":"Erdos97___HasNEquidistantPointsOn","docHtml":"\n A set of points $A$ has n equidistant points on a set of points $B$\nif for every point in $B$, there exist at least $n$ other points in $A$ that are equidistant from it.
"},"Erdos97.HasNEquidistantProperty":{"url":"/FormalConjectures/ErdosProblems/«97»/#Erdos97___HasNEquidistantProperty","anchor":"Erdos97___HasNEquidistantProperty","docHtml":"\n A set of points $A$ has n equidistant property\nif for every point in $A$, there exist at least $n$ other points in $A$ that are equidistant from it.
"},"Erdos97.HasNUnitDistancePointsAt":{"url":"/FormalConjectures/ErdosProblems/«97»/#Erdos97___HasNUnitDistancePointsAt","anchor":"Erdos97___HasNUnitDistancePointsAt","docHtml":"\n A set of points $A$ has n unit distance points at $p$\nif there exist at least $n$ other points in $A$ that are at unit distance from $p$.
"},"Erdos97.HasNUnitDistancePointsOn":{"url":"/FormalConjectures/ErdosProblems/«97»/#Erdos97___HasNUnitDistancePointsOn","anchor":"Erdos97___HasNUnitDistancePointsOn","docHtml":"\n A set of points $A$ has n unit distance points on a set of points $B$\nif for every point in $B$, there exist at least $n$ other points in $A$ that are at unit distance from it.
"},"Erdos97.HasNUnitDistanceProperty":{"url":"/FormalConjectures/ErdosProblems/«97»/#Erdos97___HasNUnitDistanceProperty","anchor":"Erdos97___HasNUnitDistanceProperty","docHtml":"\n A set of points $A$ has n unit distance property\nif for every point in $A$, there exist at least $n$ other points in $A$ that are at unit distance from it.
"},"Erdos97.erdos_97":{"url":"/FormalConjectures/ErdosProblems/«97»/#Erdos97___erdos_97","anchor":"Erdos97___erdos_97","docHtml":"\n Does every convex polygon have a vertex with no other 4 vertices equidistant from it?
"},"Erdos97.erdos_97.variants.three_equidistant":{"url":"/FormalConjectures/ErdosProblems/«97»/#Erdos97___erdos_97___variants___three_equidistant","anchor":"Erdos97___erdos_97___variants___three_equidistant","docHtml":"\n Erdős originally conjectured this (in [Er46b]) with no 3 vertices equidistant,\nbut Danzer found a convex polygon on 9 points such that every vertex has three\nvertices equidistant from it (but this distance depends on the vertex).\nDanzer's construction is explained in [Er87b].
\n\n [Er46b] Erdős, P.,
\n Erdős also conjectured that there is a $k$ for which every convex polygon has a vertex\nwith no other $k$ vertices equidistant from it.
"},"Erdos97.erdos_97.variants.three_unit_distance":{"url":"/FormalConjectures/ErdosProblems/«97»/#Erdos97___erdos_97___variants___three_unit_distance","anchor":"Erdos97___erdos_97___variants___three_unit_distance","docHtml":"\n Fishburn and Reeds [FiRe92] have found a convex polygon on 20 points such that\nevery vertex has three vertices equidistant from it (and this distance is the same for all vertices).
\n\n [FiRe92] Fishburn, P. C. and Reeds, J. A.,
\n A two-part partition ${A, B}$ of $V$ is a cut if the convex hulls of $A$ and $B$ are disjoint.
"},"Erdos97.erdos_97.variants.three_unit_distance_cut_min":{"url":"/FormalConjectures/ErdosProblems/«97»/#Erdos97___erdos_97___variants___three_unit_distance_cut_min","anchor":"Erdos97___erdos_97___variants___three_unit_distance_cut_min","docHtml":"\n Fishburn and Reeds [FiRe92] also proved that the smallest $n$ for which there exists\na convex $n$-gon and a cut ${A, B}$ of its vertices such that $|{b \\in B : d(a, b) = 1}| ≥ 3$\nfor all $a \\in A$, and $|{a \\in A : d(a, b) = 1}| ≥ 3$ for all $b \\in B$, is $n = 20$.
"},"Erdos828.erdos_828":{"url":"/FormalConjectures/ErdosProblems/«828»/#Erdos828___erdos_828","anchor":"Erdos828___erdos_828","docHtml":"\n Is it true that, for any $a \\in \\mathbb{Z}$, there are infinitely many $n$ such that\n$$\\phi(n) | n + a$$?
"},"Erdos828.erdos_828.variants.lehmer_conjecture":{"url":"/FormalConjectures/ErdosProblems/«828»/#Erdos828___erdos_828___variants___lehmer_conjecture","anchor":"Erdos828___erdos_828___variants___lehmer_conjecture","docHtml":"\n When $n > 1$, Lehmer conjectured that $\\phi(n) | n - 1$ if and only if $n$ is prime.
"},"Erdos828.erdos_828.variants.phi_dvd_self_iff_pow2_pow3":{"url":"/FormalConjectures/ErdosProblems/«828»/#Erdos828___erdos_828___variants___phi_dvd_self_iff_pow2_pow3","anchor":"Erdos828___erdos_828___variants___phi_dvd_self_iff_pow2_pow3","docHtml":"\n It is an easy exercise to show that $\\phi(n) | n$ if and only if $n = 0, 1$ or $n = 2^a 3^b$ for\nsome $a > 0$.
"},"Erdos285.erdos_285":{"url":"/FormalConjectures/ErdosProblems/«285»/#Erdos285___erdos_285","anchor":"Erdos285___erdos_285","docHtml":"\n Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1 < n_2 < \\dots < n_k$ with\n$$\n1 = \\frac{1}{n_1} + \\cdots + \\frac{1}{n_k}.\n$$\nIs it true that\n$$\nf(k) = (1 + o(1)) \\frac{e}{e - 1} k ?\n$$
\n\n Proved by Martin [Ma00].
\n\n [Ma00] Martin, Greg,
\n It is trivial that $f(k)\\geq (1 + o(1)) \\frac{e}{e - 1}k$.
"},"Erdos392.erdos_392":{"url":"/FormalConjectures/ErdosProblems/«392»/#Erdos392___erdos_392","anchor":"Erdos392___erdos_392","docHtml":"\n Let $A(n)$ denote the least value of $t$ such that\n$$\nn! = a_1 \\cdots a_t\n$$\nwith $a_1 \\leq \\cdots \\leq a_t\\leq n^2$. Then\n$$\nA(n) = \\frac{n}{2} - \\frac{n}{2\\log n} + o\\left(\\frac{n}{\\log n}\\right).\n$$
"},"Erdos392.erdos_392.variants.lower":{"url":"/FormalConjectures/ErdosProblems/«392»/#Erdos392___erdos_392___variants___lower","anchor":"Erdos392___erdos_392___variants___lower","docHtml":"\n If we change the condition to $a_t \\leq n$ it can be shown that\n$$\nA(n) = n - \\frac{n}{\\log n} + o\\left(\\frac{n}{\\log n}\\right)\n$$
"},"Erdos392.erdos_392.variants.implication":{"url":"/FormalConjectures/ErdosProblems/«392»/#Erdos392___erdos_392___variants___implication","anchor":"Erdos392___erdos_392___variants___implication","docHtml":"\n Cambie has observed that a positive answer follows from the result above with $a_t \\leq n$, simply\nby pairing variables together, e.g. taking $a'
\n Let $a$ be some sequence of natural numbers. We set $F(A,X,k)$ to be the count of\nthe number of $i$ such that $[a_i,a_{i+1}, \\dots ,a_{i+k−1}] < X$,\nwhere the left-hand side is the least common multiple.
"},"Erdos873.erdos_873":{"url":"/FormalConjectures/ErdosProblems/«873»/#Erdos873___erdos_873","anchor":"Erdos873___erdos_873","docHtml":"\n Let $A = {a_1 < a_2 < \\dots} \\subseteq \\mathbb{N}$ and let $F(A,X,k)$ count the number of $i$\nsuch that $[a_i,a_{i+1}, \\dots ,a_{i+k−1}] < X$, where the left-hand side is the least common\nmultiple. Is it true that, for every $\\epsilon > 0$, there exists some $k$ such that\n$F(A,X,k) < X^\\epsilon$?
"},"Erdos376.erdos_376":{"url":"/FormalConjectures/ErdosProblems/«376»/#Erdos376___erdos_376","anchor":"Erdos376___erdos_376","docHtml":"\n Are there infinitely many $n$ such that ${2n\\choose n}$ is coprime to $105$?
"},"Erdos376.erdos_376.variants.prime":{"url":"/FormalConjectures/ErdosProblems/«376»/#Erdos376___erdos_376___variants___prime","anchor":"Erdos376___erdos_376___variants___prime","docHtml":"\n Erdős, Graham, Ruzsa, and Straus [EGRS75] have shown that, for any two odd primes $p$ and $q$,\nthere are infinite many $n$ such that ${2n\\choose n}$ is coprime to $pq$.
"},"Erdos495.erdos_495":{"url":"/FormalConjectures/ErdosProblems/«495»/#Erdos495___erdos_495","anchor":"Erdos495___erdos_495","docHtml":"\n Let $\\alpha,\\beta \\in \\mathbb{R}$. Is it true that$$\\liminf_{n\\to \\infty} n | n\\alpha |\n| n\\beta| =0$$? This is also known as the Littlewood conjecture.
"},"Erdos89.erdos_89":{"url":"/FormalConjectures/ErdosProblems/«89»/#Erdos89___erdos_89","anchor":"Erdos89___erdos_89","docHtml":"\n Does every set of $n$ distinct points in $\\mathbb{R}^2$ determine $\\gg \\frac{n}{\\sqrt{\\log n}}$\nmany distinct distances?
"},"Erdos89.erdos_89.variants.n_dvd_log_n":{"url":"/FormalConjectures/ErdosProblems/«89»/#Erdos89___erdos_89___variants___n_dvd_log_n","anchor":"Erdos89___erdos_89___variants___n_dvd_log_n","docHtml":"\n Guth and Katz [GuKa15] proved that there are always $\\gg \\frac{n}{\\log n}$ many distinct distances.
\n\n [GuKa15] Guth, Larry and Katz, Nets Hawk, On the Erdős distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.
"},"Erdos89.erdos_89.variants.implies_n_dvd_log_n":{"url":"/FormalConjectures/ErdosProblems/«89»/#Erdos89___erdos_89___variants___implies_n_dvd_log_n","anchor":"Erdos89___erdos_89___variants___implies_n_dvd_log_n","docHtml":"\n This theorem provides a sanity check, showing that the main conjecture (erdos_89) is strictly\nstronger than the solved Guth and Katz result. It proves that, trivially, if the lower bound\n$\\frac{n}{\\sqrt{\\log n}}$ holds, then the weaker lower bound $\\frac{n}{\\log n}$ must also hold.
\n Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$\nif $(a,b)=1$ such that $\\limsup_{p,k} f(p^k) / \\log(p^k) = ∞$.\nIs it true that $\\limsup_n (f(n+1)−f(n))/ \\log n = ∞$?
\n\n The answer is no; this follows from a construction of Wirsing [Wi81], rediscovered by\nArchivara [Ar25] and formalised in Lean by Aristotle [ArWu25].
"},"Erdos897.erdos_897.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«897»/#Erdos897___erdos_897___parts___ii","anchor":"Erdos897___erdos_897___parts___ii","docHtml":"\n Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$\nif $(a,b)=1$) such that $\\limsup_{p,k} f(p^k) / \\log(p^k) = ∞$.\nIs it true that $\\limsup_n f(n+1)/ f(n) = ∞$?
\n\n The answer is no; the same counterexample is formalised in Lean by Aristotle [ArWu25].
"},"Erdos897.erdos_897.variants.log_growth":{"url":"/FormalConjectures/ErdosProblems/«897»/#Erdos897___erdos_897___variants___log_growth","anchor":"Erdos897___erdos_897___variants___log_growth","docHtml":"\n Wirsing [Wi70] proved that if $|f(n+1)−f(n)| ≤ C$ then $f(n) = c \\log n + O(1)$ for some constant\n$c$.
"},"Erdos897.erdos_897.variants.parts.i":{"url":"/FormalConjectures/ErdosProblems/«897»/#Erdos897___erdos_897___variants___parts___i","anchor":"Erdos897___erdos_897___variants___parts___i","docHtml":"\n Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$\nif $(a,b)=1$) such that $\\limsup_{p,k} f(p^k) / \\log(p^k) = ∞$ and $f(p^k) = f(p)$\nor $f(p^k) = kf(p)$.\nIs it true that $\\limsup_n (f(n+1)−f(n))/ \\log n = ∞$?
\n\n The known counterexample does not satisfy either of these extra hypotheses, so this variant remains\nopen.
"},"Erdos897.erdos_897.variants.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«897»/#Erdos897___erdos_897___variants___parts___ii","anchor":"Erdos897___erdos_897___variants___parts___ii","docHtml":"\n Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$\nif $(a,b)=1$) such that $\\limsup_{p,k} f(p^k) / \\log(p^k) = ∞$ and $f(p^k) = f(p)$\nor $f(p^k) = kf(p)$.\nIs it true that $\\limsup_n f(n+1)/f(n) = ∞$?
\n\n The known counterexample does not satisfy either of these extra hypotheses, so this variant remains\nopen.
"},"Erdos195.erdos_195":{"url":"/FormalConjectures/ErdosProblems/«195»/#Erdos195___erdos_195","anchor":"Erdos195___erdos_195","docHtml":"\n What is the largest $k$ such that in any permutation of $\\mathbb{Z}$ there must exist a\nmonotone $k$-term arithmetic progression $x_1 < \\cdots < x_k$?
"},"Erdos195.erdos_195.variants.leq_5_bound":{"url":"/FormalConjectures/ErdosProblems/«195»/#Erdos195___erdos_195___variants___leq_5_bound","anchor":"Erdos195___erdos_195___variants___leq_5_bound","docHtml":"\n Geneson [Ge19] proved that k ≤ 5.
"},"Erdos195.erdos_195.variants.leq_4_bound":{"url":"/FormalConjectures/ErdosProblems/«195»/#Erdos195___erdos_195___variants___leq_4_bound","anchor":"Erdos195___erdos_195___variants___leq_4_bound","docHtml":"\n Adenwalla [Ad22] proved that k ≤ 4.
"},"Erdos7.erdos_7":{"url":"/FormalConjectures/ErdosProblems/«7»/#Erdos7___erdos_7","anchor":"Erdos7___erdos_7","docHtml":"\n Is there a covering system all of whose moduli are odd (and greater than 1)?
"},"Erdos244.erdos_244":{"url":"/FormalConjectures/ErdosProblems/«244»/#Erdos244___erdos_244","anchor":"Erdos244___erdos_244","docHtml":"\n Let $C > 1$. Does the set of integers of the form $p + \\lfloor C^k \\rfloor$,\nfor some prime $p$ and $k\\geq 0$, have density $>0$?
"},"Erdos244.erdos_244.variants.Romanoff":{"url":"/FormalConjectures/ErdosProblems/«244»/#Erdos244___erdos_244___variants___Romanoff","anchor":"Erdos244___erdos_244___variants___Romanoff","docHtml":"\n Romanoff [Ro34] proved that the answer is yes if $C$ is an integer.
\n\n [Ro34] Romanoff, N. P.,
\n $Q_3$ is the 3-dimensional hypercube graph (8 vertices, 12 edges).\nVertices are 3-bit vectors. Two vertices are adjacent iff they differ in exactly one bit.
"},"Erdos567.K33":{"url":"/FormalConjectures/ErdosProblems/«567»/#Erdos567___K33","anchor":"Erdos567___K33","docHtml":"\n $K_{3,3}$ is the complete bipartite graph with partition sizes 3, 3 (6 vertices, 9 edges).
"},"Erdos567.H5":{"url":"/FormalConjectures/ErdosProblems/«567»/#Erdos567___H5","anchor":"Erdos567___H5","docHtml":"\n $H_5$ is $C_5$ with two vertex-disjoint chords (5 vertices, 7 edges).\nAlso known as $K_4^*$ (the graph obtained from $K_4$ by subdividing one edge).
"},"Erdos567.erdos_567.parts.i":{"url":"/FormalConjectures/ErdosProblems/«567»/#Erdos567___erdos_567___parts___i","anchor":"Erdos567___erdos_567___parts___i","docHtml":"\nErdős Problem 567 (Q3)
\n\n Is $Q_3$ (the 3-dimensional hypercube) Ramsey size linear?
"},"Erdos567.erdos_567.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«567»/#Erdos567___erdos_567___parts___ii","anchor":"Erdos567___erdos_567___parts___ii","docHtml":"\nErdős Problem 567 (K33)
\n\n Is $K_{3,3}$ Ramsey size linear?
"},"Erdos567.erdos_567.parts.iii":{"url":"/FormalConjectures/ErdosProblems/«567»/#Erdos567___erdos_567___parts___iii","anchor":"Erdos567___erdos_567___parts___iii","docHtml":"\nErdős Problem 567 (H5)
\n\n Is $H_5$ ($C_5$ with two vertex-disjoint chords) Ramsey size linear?
"},"Erdos64.erdos_64":{"url":"/FormalConjectures/ErdosProblems/«64»/#Erdos64___erdos_64","anchor":"Erdos64___erdos_64","docHtml":"\n Does every finite graph with minimum degree at least $3$\ncontain a cycle of length $2^k$ for some $k \\geq 2$?
"},"Erdos434.Nat.IsRepresentableAs":{"url":"/FormalConjectures/ErdosProblems/«434»/#Erdos434___Nat___IsRepresentableAs","anchor":"Erdos434___Nat___IsRepresentableAs","docHtml":"\n A natural $n$ is representable as a set $A$ if it can be\nwritten as the sum of finitely many elements of $A$\n(with repetition allowed).
"},"Erdos434.Nat.NcardUnrepresentable":{"url":"/FormalConjectures/ErdosProblems/«434»/#Erdos434___Nat___NcardUnrepresentable","anchor":"Erdos434___Nat___NcardUnrepresentable","docHtml":"\n The number of naturals that cannot be written as the sum of\nfinitely many elements of the set $A$, with repetition allowed.
"},"Erdos434.erdos_434.parts.i":{"url":"/FormalConjectures/ErdosProblems/«434»/#Erdos434___erdos_434___parts___i","anchor":"Erdos434___erdos_434___parts___i","docHtml":"\n Let $k \\le n$. What choice of $A\\subseteq{1, \\dots, n}$ (with $\\text{gcd}(A) = 1$) of size $|A| = k$\nmaximises the number of integers not representable as the sum of finitely\nmany elements from $A$ (with repetitions allowed)?\nIs it ${n, n - 1, \\dots, n - k + 1}$?
"},"Erdos434.erdos_434.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«434»/#Erdos434___erdos_434___parts___ii","anchor":"Erdos434___erdos_434___parts___ii","docHtml":"\n Let $k \\le n$. Out of all $A\\subseteq{1, \\dots, n}$ (with $\\text{gcd}(A) = 1$) of size $|A| = k$,\ndoes $A = {n, n - 1, \\dots, n - k + 1}$ maximise the number of integers\nnot representable as the sum of finitely many elements from $A$ (with repetitions allowed)?
"},"Erdos1106.p":{"url":"/FormalConjectures/ErdosProblems/«1106»/#Erdos1106___p","anchor":"Erdos1106___p","docHtml":"\n The partition function p(n) is the number of ways to write n as a sum of positive\nintegers (where the order of the summands does not matter).
"},"Erdos1106.erdos_1106.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1106»/#Erdos1106___erdos_1106___parts___i","anchor":"Erdos1106___erdos_1106___parts___i","docHtml":"\n Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of\n$∏_{i= 1} ^ {n} p(n)$, then $F(n)$ tends to infinity when $n$ tends to infinity.
"},"Erdos1106.erdos_1106.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«1106»/#Erdos1106___erdos_1106___parts___ii","anchor":"Erdos1106___erdos_1106___parts___ii","docHtml":"\n Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of\n$∏_{i= 1} ^ {n} p(n)$, $F(n)>n$ for sufficiently large $n$.
"},"Erdos38.erdos_38":{"url":"/FormalConjectures/ErdosProblems/«38»/#Erdos38___erdos_38","anchor":"Erdos38___erdos_38","docHtml":"\n Does there exist $B \\subset \\mathbb{N}$ which is not an additive basis,\nbut is such that for every set $A \\subseteq \\mathbb{N}$ of Schnirelmann density $\\alpha$\nand every $N$ there exists $b \\in B$ such that\n$$\n\\lvert (A \\cup (A+b)) \\cap {1, \\ldots, N} \\rvert \\geq (\\alpha + f(\\alpha)) N\n$$\nwhere $f(\\alpha) > 0$ for $0 < \\alpha < 1$?
\n\n Note: here Erdős seems to use a slightly weaker notion of an additive basis (see [Er56] at the top\nof page 135). In particular, for this problem, a set is an additive basis of order $k$ if every\nnatural number can be written as a sum of
\n A positive solution was given by GPT 5.5 Pro\n(prompted by gebyjaff, cleanup by Liam Price); in fact a sparse random set $B$ has this property,\nwith $f(\\alpha)\\gg \\alpha (1-\\alpha)^2$.
"},"Erdos835.Property":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___Property","anchor":"Erdos835___Property","docHtml":"\n The property that for a given $k$, the $k$-subsets of a $2k$-set can be colored with $k+1$ colors\nsuch that any $(k+1)$-subset contains all colors.
"},"Erdos835.erdos_835":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___erdos_835","anchor":"Erdos835___erdos_835","docHtml":"\n Does there exist a $k>2$ such that the $k$-sized subsets of {1,...,2k} can be coloured with\n$k+1$ colours such that for every $A\\subset {1,\\ldots,2k}$ with $\\lvert A\\rvert=k+1$ all $k+1$\ncolours appear among the $k$-sized subsets of $A$?
"},"Erdos835.property_iff_chromaticNumber":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___property_iff_chromaticNumber","anchor":"Erdos835___property_iff_chromaticNumber"},"Erdos835.erdos_835.variants.johnson":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___erdos_835___variants___johnson","anchor":"Erdos835___erdos_835___variants___johnson","docHtml":"\n Alternative statement of Erdős Problem 835 using the chromatic number of the Johnson graph.\nThis is equivalent to asking whether there exists $k > 2$ such that the chromatic number of the\nJohnson graph $J(2k, k)$ is $k+1$.
"},"Erdos835.johnsonGraph_2k_k_chromaticNumber_known_cases":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___johnsonGraph_2k_k_chromaticNumber_known_cases","anchor":"Erdos835___johnsonGraph_2k_k_chromaticNumber_known_cases","docHtml":"\n It is known that for $3 \\leq k \\leq 8$, the chromatic number of $J(2k, k)$ is greater than $k+1$,\nsee Johnson graphs.
"},"Erdos835.johnsonGraph_18_9_chromaticNumber":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___johnsonGraph_18_9_chromaticNumber","anchor":"Erdos835___johnsonGraph_18_9_chromaticNumber","docHtml":"\n The smallest case not on this page is $k=9$:\nBut that one can be solved as well:\nThe chromatic number of $J(18, 9)$ is at least $11$.
"},"Erdos835.johnsonBound":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___johnsonBound","anchor":"Erdos835___johnsonBound","docHtml":"\n Johnson's upper bound on the maximum size A(n, d, w) of a n-dimensional binary code of\ndistance d and weight w is as follows:
\n If d > 2 * w, then A(n, d, w) = 1.
\n If d ≤ 2 * w, then A(n, d, w) ≤ ⌊n / w * A(n - 1, d, w - 1)⌋.
\n Johnson's bound for the independence number of the Johnson graph.
"},"Erdos835.div_johnsonBound_le_chromaticNum_johnson":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___div_johnsonBound_le_chromaticNum_johnson","anchor":"Erdos835___div_johnsonBound_le_chromaticNum_johnson","docHtml":"\n Johnson's bound for the chromatic number of the Johnson graph.
"},"Erdos835.chromaticNumber_johnson_2k_k_lower_bound":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___chromaticNumber_johnson_2k_k_lower_bound","anchor":"Erdos835___chromaticNumber_johnson_2k_k_lower_bound","docHtml":"\n It is known that for $3 \\leq k \\leq 8$, the chromatic number of $J(2k, k)$ is greater than\n$k+1$, see Johnson graphs.
"},"Erdos835.chromaticNumber_johnson_2k_k_lower_bound_odd":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___chromaticNumber_johnson_2k_k_lower_bound_odd","anchor":"Erdos835___chromaticNumber_johnson_2k_k_lower_bound_odd","docHtml":"\n It is also known that for $3 \\leq k \\leq 203$ odd, the chromatic number of $J(2k, k)$ is\ngreater than $k+1$, see Johnson graphs.
"},"Erdos835.johnson_chromaticNumber_odd":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___johnson_chromaticNumber_odd","anchor":"Erdos835___johnson_chromaticNumber_odd","docHtml":"\n It can be seen that the chromatic number of $J(2k,k)$ is $>k+1$ for all odd $k>2$.
"},"Erdos835.johnson_chromaticNumber_composite":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___johnson_chromaticNumber_composite","anchor":"Erdos835___johnson_chromaticNumber_composite","docHtml":"\n Ma and Tang have proved that the chromatic number of $J(2k,k)$ is $>k+1$ for all $k>2$ not of the\nform $p-1$ for prime $p$.
"},"Erdos835.johnsonGraph_chromaticNumber_odd_of_johnson_chromaticNumber_composite":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___johnsonGraph_chromaticNumber_odd_of_johnson_chromaticNumber_composite","anchor":"Erdos835___johnsonGraph_chromaticNumber_odd_of_johnson_chromaticNumber_composite","docHtml":"\n Ma and Tang's result implies the cases for odd $k$.
"},"Erdos835.johnson_chromaticNumber":{"url":"/FormalConjectures/ErdosProblems/«835»/#Erdos835___johnson_chromaticNumber","anchor":"Erdos835___johnson_chromaticNumber","docHtml":"\n Is the chromatic number of J(2 * k, k) always at least k + 2?
\n Let $f(n)$ be the smallest integer for which every graph on $n$ vertices with minimal degree $\\geq\nf(n)$ contains a $C_4$.
"},"Erdos85.erdos_85":{"url":"/FormalConjectures/ErdosProblems/«85»/#Erdos85___erdos_85","anchor":"Erdos85___erdos_85","docHtml":"\n Is it true that, for all large $n$, $f(n + 1) \\ge f(n)$?
"},"Erdos1175.erdos_1175":{"url":"/FormalConjectures/ErdosProblems/«1175»/#Erdos1175___erdos_1175","anchor":"Erdos1175___erdos_1175","docHtml":"\n Let $\\kappa$ be an uncountable cardinal. Must there exist a cardinal $\\lambda$ such that every\ngraph with chromatic number $\\lambda$ contains a triangle-free subgraph with chromatic number\n$\\kappa$?
\n\n Shelah proved that a negative answer is consistent when\n$\\kappa = \\lambda = \\aleph_1$ (see erdos_1175.variants.shelah_consistency).
\nShelah's consistency result: it is consistent with ZFC that there exists a graph $G$ with\nchromatic number $\\aleph_1$ such that every triangle-free subgraph of $G$ has chromatic number\nstrictly less than $\\aleph_1$.
\n\n This shows that a negative answer to Problem 1175 (with $\\kappa = \\lambda = \\aleph_1$) is\nconsistent, so the main statement erdos_1175 is not provable in ZFC.
\nFormalization caveat (consistency placeholder). Shelah's result is a answer(sorry) consistency placeholder: the\nintended conjecture is the model-theoretic statement, and any concrete formalisation must\neither appeal to an explicit extra axiom (such as Shelah's specific forcing extension)\nor to a meta-theoretic consistency proof. Until such a wrapper exists in FormalConjectures,\nwe leave the body as sorry.
\nThreshold reformulation variant. Replaces chromaticCardinal = λ in the hypothesis\nof erdos_1175 with λ ≤ chromaticCardinal (a graph of chromatic number ≥ λ has a\ntriangle-free subgraph of chromatic number κ). This is a strengthening of erdos_1175\n(see erdos_1175.test.threshold_implies_exact).
\n Every graph has a triangle-free subgraph: the bottom subgraph (with no edges)\nwitnesses triangle-freeness, so the existential\n∃ H : G.Subgraph, H.coe.CliqueFree 3 in erdos_1175 is non-vacuous.
\n The threshold variant threshold_formulation is stronger than the exact-equality\nform erdos_1175: if every graph with chromaticCardinal ≥ μ has the desired\ntriangle-free subgraph, then in particular every graph with chromaticCardinal = μ\ndoes too.
\n Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of\ndegree $n$ with coefficients in ${-1, 1}$,\n$$\\max_{|z|=1} |P(z)| > (1 + c) \\sqrt{n}?$$
"},"Erdos1150.erdos_1150.variants.parseval_lower_bound":{"url":"/FormalConjectures/ErdosProblems/«1150»/#Erdos1150___erdos_1150___variants___parseval_lower_bound","anchor":"Erdos1150___erdos_1150___variants___parseval_lower_bound","docHtml":"\n The trivial lower bound from Parseval's identity: for any polynomial $P$ of degree $n$ with\ncoefficients in ${-1, 1}$, we have $\\max_{|z|=1} |P(z)| \\geq \\sqrt{n+1}$.
\n\n This follows from Parseval's identity:\n$$\\frac{1}{2\\pi} \\int_0^{2\\pi} |P(e^{i\\theta})|^2 d\\theta = \\sum_{k=0}^{n} |a_k|^2 = n+1$$\nsince each $|a_k|^2 = 1$.
"},"Erdos1062.ForkFree":{"url":"/FormalConjectures/ErdosProblems/«1062»/#Erdos1062___ForkFree","anchor":"Erdos1062___ForkFree","docHtml":"\n A set A of positive integers is fork-free if no element divides two distinct\nother elements of A.
\n The extremal function from Erdős problem 1062: the largest size of a fork-free subset of\n{1,...,n}.
\n Erdős asked whether the limiting density f n / n exists and, if so, whether it is\nirrational.
\n The interval [⌊n/3⌋, n] is fork-free, and therefore f n is at least ⌈2n / 3⌉.
\n Lebensold proved that for large n, the function f n lies between 0.6725 n and\n0.6736 n.
\n
\n [Ta26] T. Tao, Erdős problem 358 (2026)
\n\n When $A_n = n$, the function $f$ defined above counts the number of odd divisors of $n$.
"},"Erdos358.erdos_358.parts.i":{"url":"/FormalConjectures/ErdosProblems/«358»/#Erdos358___erdos_358___parts___i","anchor":"Erdos358___erdos_358___parts___i","docHtml":"\n Let $A={a_1 < \\cdots}$ be an infinite sequence of integers. Let $f(n)$ count the number of\nsolutions to $$n=\\sum_{u\\leq i\\leq v}a_i.$$\nIs there such an $A$ for which $f(n)\\to \\infty$ as $n\\to \\infty$?
\n\n Tao [Ta26] constructed such a sequence with $f(n) \\gg \\log n$ for all sufficiently large $n$.
"},"Erdos358.erdos_358.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«358»/#Erdos358___erdos_358___parts___ii","anchor":"Erdos358___erdos_358___parts___ii","docHtml":"\n Let $A={a_1 < \\cdots}$ be an infinite sequence of integers. Let $f(n)$ count the number of\nsolutions to $$n=\\sum_{u\\leq i\\leq v}a_i.$$\nIs there an $A$ such that $f(n)\\geq 2$ for all large $n$?
\n\n This also follows from Tao's construction with $f(n) \\gg \\log n$ [Ta26].
"},"Erdos358.erdos_358.variants.prime_set":{"url":"/FormalConjectures/ErdosProblems/«358»/#Erdos358___erdos_358___variants___prime_set","anchor":"Erdos358___erdos_358___variants___prime_set","docHtml":"\n When $A ={a_1 < \\cdots}$ corresponds to the set of primes, it is conjectured that the\n$\\limsup$ of the number of representations $$n=\\sum_{u\\leq i\\leq v}a_i$$ is infinite.
"},"Erdos358.erdos_358.variants.prime_set_density_representation":{"url":"/FormalConjectures/ErdosProblems/«358»/#Erdos358___erdos_358___variants___prime_set_density_representation","anchor":"Erdos358___erdos_358___variants___prime_set_density_representation","docHtml":"\n When $A ={a_1 < \\cdots}$ corresponds to the set of primes, it is conjectured that the set of\nnumbers $n$ that have representations $$n=\\sum_{u\\leq i\\leq v}a_i$$ has positive upper density.
"},"Erdos358.erdos_358.variants.one_le":{"url":"/FormalConjectures/ErdosProblems/«358»/#Erdos358___erdos_358___variants___one_le","anchor":"Erdos358___erdos_358___variants___one_le","docHtml":"\n It is conjectured that if $A ={a_1 < \\cdots}$ and $g$ counts the number of representations\n$$n=\\sum_{u\\leq i\\leq v}a_i$$ such that the sum has at least two terms, then for all $n$ we have\n$1 \\leq g(n)$ for sufficiently large $n$.
"},"Erdos351.imageSet":{"url":"/FormalConjectures/ErdosProblems/«351»/#Erdos351___imageSet","anchor":"Erdos351___imageSet","docHtml":"\n The set of rational numbers of the form P(n) + 1 / n where n is a natural number\nand P is a polynomial with rational coefficients.
\n Note: We include P 0 in there (since 1 / 0 = 0), but this doesn't change the validity of the\nconjecture
\n The predicate that a set A is strongly complete, i.e. that for every finite set B, every sufficiently\nlarge integer is a sum of elements of the set A \\ B.
\n The predicate that the rational polynomial P has a complete image.
\n Let $p(x) \\in \\mathbb{Q}[x]$ be a non-constant rational polynomial with positive leading\ncoefficient. Is it true that $$A={ p(n)+1/n : n \\in \\mathbb{N}}$$ is strongly complete,\nin the sense that, for any finite set $B$,\n$$\\left{\\sum_{a \\in X} a : X \\subseteq A \\setminus B, X \\textrm{ is finite}\\right}$$\ncontains all sufficiently large integers?
"},"Erdos351.erdos_351.variants.X":{"url":"/FormalConjectures/ErdosProblems/«351»/#Erdos351___erdos_351___variants___X","anchor":"Erdos351___erdos_351___variants___X","docHtml":"\n Let $p(x) = x \\in \\mathbb{Q}[x]$. It has been shown that\n$$A={ p(n)+1/n : n \\in \\mathbb{N}}$$\nis strongly complete, in the sense that, for any finite set $B$,\n$$\\left{\\sum_{a \\in X} a : X \\subseteq A \\setminus B, X \\textrm{ is finite}\\right}$$\ncontains all sufficiently large integers.
"},"Erdos351.erdos_351.variants.X_sq":{"url":"/FormalConjectures/ErdosProblems/«351»/#Erdos351___erdos_351___variants___X_sq","anchor":"Erdos351___erdos_351___variants___X_sq","docHtml":"\n Let $p(x) = x ^ 2 \\in \\mathbb{Q}[x]$. It has been shown that\n$$A={ p(n)+1/n : n \\in \\mathbb{N}}$$\nis strongly complete, in the sense that, for any finite set $B$,\n$$\\left{\\sum_{a \\in X} a : X \\subseteq A \\setminus B, X \\textrm{ is finite}\\right}$$\ncontains all sufficiently large integers.
"},"Erdos686.erdos_686":{"url":"/FormalConjectures/ErdosProblems/«686»/#Erdos686___erdos_686","anchor":"Erdos686___erdos_686","docHtml":"\n Can every integer $N≥2$ be written as\n$$N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?
"},"Erdos686.erdos_686.variants.square":{"url":"/FormalConjectures/ErdosProblems/«686»/#Erdos686___erdos_686___variants___square","anchor":"Erdos686___erdos_686___variants___square","docHtml":"\n Can every square $N≥2$ be written as\n$$N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?
"},"Erdos686.erdos_686.variants.four":{"url":"/FormalConjectures/ErdosProblems/«686»/#Erdos686___erdos_686___variants___four","anchor":"Erdos686___erdos_686___variants___four","docHtml":"\n Can $4$ be written as\n$$4=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?
"},"Erdos686.erdos_686.variants.four_two":{"url":"/FormalConjectures/ErdosProblems/«686»/#Erdos686___erdos_686___variants___four_two","anchor":"Erdos686___erdos_686___variants___four_two","docHtml":"\n The number $4$ cannot be written as\n$$4=\\frac{\\prod_{1\\leq i\\leq 2}(m+i)}{\\prod_{1\\leq i\\leq 2}(n+i)}$$\nfor $m≥n+2$!
"},"Erdos686.erdos_686.variants.four_three":{"url":"/FormalConjectures/ErdosProblems/«686»/#Erdos686___erdos_686___variants___four_three","anchor":"Erdos686___erdos_686___variants___four_three","docHtml":"\n The number $4$ cannot be written as\n$$4=\\frac{\\prod_{1\\leq i\\leq 2}(m+i)}{\\prod_{1\\leq i\\leq 2}(n+i)}$$\nfor $m≥n+2$!
\n\n See comment section on erdosproblems.com
"},"Erdos686.erdos_686.variants.nine":{"url":"/FormalConjectures/ErdosProblems/«686»/#Erdos686___erdos_686___variants___nine","anchor":"Erdos686___erdos_686___variants___nine","docHtml":"\n Can $9$ be written as\n$$9=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?
"},"Erdos686.erdos_686.variants.twenty_five":{"url":"/FormalConjectures/ErdosProblems/«686»/#Erdos686___erdos_686___variants___twenty_five","anchor":"Erdos686___erdos_686___variants___twenty_five","docHtml":"\n Can $25$ be written as\n$$25=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?
"},"Erdos686.erdos_686.variants.non_square":{"url":"/FormalConjectures/ErdosProblems/«686»/#Erdos686___erdos_686___variants___non_square","anchor":"Erdos686___erdos_686___variants___non_square","docHtml":"\n Can every non-square $N≥2$ be written as\n$$N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?
"},"Erdos951.Erdos951Prop":{"url":"/FormalConjectures/ErdosProblems/«951»/#Erdos951___Erdos951Prop","anchor":"Erdos951___Erdos951Prop","docHtml":"\n A sequence a : ℕ → ℝ is said to have property Erdos951Prop if for any pair of distinct\nfinitely supported sequences k l : ℕ →₀ ℕ their corresponding Beurling integers are of distance\nat least one apart.
\n If a has property Erdos951Prop and 1 < a 0, then a is a set of Beurling\nprime numbers.
\n If 1 < a 0 < ... has property Erdos951Prop, is it true that #{a i ≤ x} ≤ π x?
\n Beurling conjectured that if the number of Beurling integer in [1, x]\nis x + o(log x), then a must be the sequence of primes.
\n Let c₁, c₂ > 0. Is it true that for any sufficiently large x, there exists more than\nc₁ * log x many consecutive primes ≤ x such that the difference between any two is > c₂?
\n It is well-known that the conjecture above is true when c₁ is sufficiently small.
\n Let $s_1 < s_2 < \\cdots$ be the sequence of squarefree numbers.
"},"Erdos145.A":{"url":"/FormalConjectures/ErdosProblems/«145»/#Erdos145___A","anchor":"Erdos145___A","docHtml":"\n Let $A(x)$ denote the set of indices $n$ for which $s_n \\leq x$.
"},"Erdos145.erdos_145":{"url":"/FormalConjectures/ErdosProblems/«145»/#Erdos145___erdos_145","anchor":"Erdos145___erdos_145","docHtml":"\n Let $s_1 < s_2 < \\cdots$ be the sequence of squarefree numbers. Is it true that, for any\n$\\alpha\\geq 0$,\n$$\n\\lim_{x\\to\\infty} \\frac{1}{x}\\sum_{s_n\\leq x}(s_{n+1}-s_n)^\\alpha\n$$\nexists?
"},"Erdos145.erdos_145.variants.le_two":{"url":"/FormalConjectures/ErdosProblems/«145»/#Erdos145___erdos_145___variants___le_two","anchor":"Erdos145___erdos_145___variants___le_two","docHtml":"\n Erdős [Er51] proved this for all $0\\leq \\alpha\\leq 2$.
\n\n [Er51] Erdös, P., Some problems and results in elementary number theory.\nPubl. Math. Debrecen (1951), 103-109.
"},"Erdos145.erdos_145.variants.le_three":{"url":"/FormalConjectures/ErdosProblems/«145»/#Erdos145___erdos_145___variants___le_three","anchor":"Erdos145___erdos_145___variants___le_three","docHtml":"\n Hooley [Ho73] extended this to all $0 \\leq \\alpha\\leq 3$.
\n\n [Ho73] Hooley, Christopher, On the intervals between consecutive terms of sequences. Proc. Symp. Pure Math, vol. 24, pp. 129-140. 1973.
"},"Erdos145.erdos_145.variants.le_eleven_thirds":{"url":"/FormalConjectures/ErdosProblems/«145»/#Erdos145___erdos_145___variants___le_eleven_thirds","anchor":"Erdos145___erdos_145___variants___le_eleven_thirds","docHtml":"\n Greaves, Harman, and Huxley [GHH97] showed that this is true for $0 \\leq \\alpha\\leq 11/3$.
\n\n [GHH97] Greaves, G. R. H. and Harman, G. and Huxley, M. N., Sieve Methods, Exponential Sums, and\ntheir Applications in Number Theory. (1997).
"},"Erdos361.erdos_361.bigO":{"url":"/FormalConjectures/ErdosProblems/«361»/#Erdos361___erdos_361___bigO","anchor":"Erdos361___erdos_361___bigO","docHtml":"\n Let $c > 0$ and $n$ be some large integer. What is the size of the largest set\n$A \\subseteq {1, \\ldots, \\lfloor c n \\rfloor}$ such that $n$ is not a sum of a subset of $A$?\nDoes this depend on $n$ in an irregular way?
"},"Erdos361.erdos_361.bigTheta":{"url":"/FormalConjectures/ErdosProblems/«361»/#Erdos361___erdos_361___bigTheta","anchor":"Erdos361___erdos_361___bigTheta","docHtml":"\n Let $c > 0$ and $n$ be some large integer. What is the size of the largest set\n$A \\subseteq {1, \\ldots, \\lfloor c n \\rfloor}$ such that $n$ is not a sum of a subset of $A$?\nDoes this depend on $n$ in an irregular way?
"},"Erdos361.erdos_361.smallO":{"url":"/FormalConjectures/ErdosProblems/«361»/#Erdos361___erdos_361___smallO","anchor":"Erdos361___erdos_361___smallO","docHtml":"\n Let $c > 0$ and $n$ be some large integer. What is the size of the largest set\n$A \\subseteq {1, \\ldots, \\lfloor c n \\rfloor}$ such that $n$ is not a sum of a subset of $A$?\nDoes this depend on $n$ in an irregular way?
"},"Erdos15.erdos_15":{"url":"/FormalConjectures/ErdosProblems/«15»/#Erdos15___erdos_15","anchor":"Erdos15___erdos_15","docHtml":"\n Is it true that $\\sum_{n=1}^\\infty(-1)^n\\frac{n}{p_n}$ converges,\nwhere $p_n$ is the sequence of primes?
\n\n Note: In the problem statement, $p_n$ is the $n$-th prime, indexed such that $p_1=2, p_2=3, \\ldots$.\nWe 0-index here to reflect how Nat.nth works.
"},"Erdos9.Erdos9A":{"url":"/FormalConjectures/ErdosProblems/«9»/#Erdos9___Erdos9A","anchor":"Erdos9___Erdos9A","docHtml":"\n The set of odd numbers that cannot be expressed as a prime plus two powers of 2.
"},"Erdos9.erdos9A_contains_one":{"url":"/FormalConjectures/ErdosProblems/«9»/#Erdos9___erdos9A_contains_one","anchor":"Erdos9___erdos9A_contains_one"},"Erdos9.erdos9A_contains_three":{"url":"/FormalConjectures/ErdosProblems/«9»/#Erdos9___erdos9A_contains_three","anchor":"Erdos9___erdos9A_contains_three"},"Erdos9.erdos9A_not_contains_five":{"url":"/FormalConjectures/ErdosProblems/«9»/#Erdos9___erdos9A_not_contains_five","anchor":"Erdos9___erdos9A_not_contains_five"},"Erdos9.erdos_9.variants.infinite":{"url":"/FormalConjectures/ErdosProblems/«9»/#Erdos9___erdos_9___variants___infinite","anchor":"Erdos9___erdos_9___variants___infinite","docHtml":"\n The set is known to be infinite. In [Er77c] Erdős credits Schinzel with proving that there are\ninfinitely many odd integers not of this form, but gives no reference.
\n\n [Er77c] Erdős, P.,
\n Is the upper density of the set of odd numbers that cannot be expressed as a prime plus\ntwo powers of 2 positive?
"},"Erdos213.Erdos213For":{"url":"/FormalConjectures/ErdosProblems/«213»/#Erdos213___Erdos213For","anchor":"Erdos213___Erdos213For","docHtml":"\n The predicate (on $n$) that there exist $n$ points in $\\mathbb{R}^2$,\nno three on a line and no four on a circle,\nsuch that all pairwise distances are integers.
"},"Erdos213.erdos_213":{"url":"/FormalConjectures/ErdosProblems/«213»/#Erdos213___erdos_213","anchor":"Erdos213___erdos_213","docHtml":"\n Let $n \\geq 4$. Are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle,\nsuch that all pairwise distances are integers?
"},"Erdos213.erdos_213.variants.KK08":{"url":"/FormalConjectures/ErdosProblems/«213»/#Erdos213___erdos_213___variants___KK08","anchor":"Erdos213___erdos_213___variants___KK08","docHtml":"\n The best construction to date, due to Kreisel and Kurz, has $n = 7$.
"},"Erdos400.g":{"url":"/FormalConjectures/ErdosProblems/«400»/#Erdos400___g","anchor":"Erdos400___g","docHtml":"\n For any $k\\geq 2$ let $g_k(n)$ denote the maximum value of $(a_1+\\cdots+a_k)-n$\nwhere $a_1,\\ldots,a_k$ are integers such that $a_1!\\cdots a_k! \\mid n!$.
"},"Erdos400.erdos_400.parts.i":{"url":"/FormalConjectures/ErdosProblems/«400»/#Erdos400___erdos_400___parts___i","anchor":"Erdos400___erdos_400___parts___i","docHtml":"\n Can one show that $\\sum_{n\\leq x}g_k(n) \\sim c_k x\\log x$ for some constant $c_k$?
"},"Erdos400.erdos_400.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«400»/#Erdos400___erdos_400___parts___ii","anchor":"Erdos400___erdos_400___parts___ii","docHtml":"\n Is it true that there is a constant $c_k$ such that for almost all $n < x$ we have\n$g_k(n)=c_k\\log x+o(\\log x)$?
"},"Erdos400.erdos_400.variants.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«400»/#Erdos400___erdos_400___variants___upper_bound","anchor":"Erdos400___erdos_400___variants___upper_bound","docHtml":"\n Erdős and Graham write that it is easy to show that $g_k(n) \\ll_k \\log n$ always, but the best\npossible constant is unknown.
"},"Erdos400.erdos_400.variants.g_pos":{"url":"/FormalConjectures/ErdosProblems/«400»/#Erdos400___erdos_400___variants___g_pos","anchor":"Erdos400___erdos_400___variants___g_pos","docHtml":"\n For $k \\ge 2$, $g_k(n) > 0$. We show this by choosing $a = (n, 1, 0, \\ldots, 0)$.
"},"Erdos196.erdos_196":{"url":"/FormalConjectures/ErdosProblems/«196»/#Erdos196___erdos_196","anchor":"Erdos196___erdos_196","docHtml":"\n Must every permutation of $\\mathbb{N}$, contain a monotone 4-term arithmetic progression?
"},"Erdos480.erdos_480":{"url":"/FormalConjectures/ErdosProblems/«480»/#Erdos480___erdos_480","anchor":"Erdos480___erdos_480","docHtml":"\n Let $x_1,x_2,\\ldots\\in [0,1]$ be an infinite sequence.\nIs it true that\n$$\\inf_n \\liminf_{m\\to \\infty} n \\lvert x_{m+n}-x_m\\rvert\\leq 5^{-1/2}\\approx 0.447?$$\nA conjecture of Newman.
"},"Erdos480.erdos_480.variants.chung_graham":{"url":"/FormalConjectures/ErdosProblems/«480»/#Erdos480___erdos_480___variants___chung_graham","anchor":"Erdos480___erdos_480___variants___chung_graham","docHtml":"\n This was proved by Chung and Graham \\cite{ChGr84}, who in fact prove that\n$$\\inf_n \\liminf_{m\\to \\infty} n \\lvert x_{m+n}-x_m\\rvert\\leq \\frac{1}{c}\\approx 0.3944$$\nwhere\n$$c=1+\\sum_{k\\geq 1}\\frac{1}{F_{2k}}=2.5353705\\cdots$$\nand $F_m$ is the $m$th Fibonacci number.
"},"Erdos480.erdos_480.variants.chung_graham_best_possible":{"url":"/FormalConjectures/ErdosProblems/«480»/#Erdos480___erdos_480___variants___chung_graham_best_possible","anchor":"Erdos480___erdos_480___variants___chung_graham_best_possible","docHtml":"\n They also prove that this constant is best possible.
"},"Erdos592.erdos_592":{"url":"/FormalConjectures/ErdosProblems/«592»/#Erdos592___erdos_592","anchor":"Erdos592___erdos_592","docHtml":"\n Determine which countable ordinals $β$ have the property that, if $α = \\omega^β$, then in any\nred/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.
"},"Erdos939.Erdos939Sums":{"url":"/FormalConjectures/ErdosProblems/«939»/#Erdos939___Erdos939Sums","anchor":"Erdos939___Erdos939Sums","docHtml":"\n A set S belongs to Erdos939Sums r if it meets the following criteria:
\n The size of the set is $|S| = r - 2$.
\n The elements of the set are coprime (their greatest common divisor is 1).
\n\n Every element in S is an $r$-powerful number.
\n The sum of the elements in S, i.e., $\\sum_{s \\in S} s$, is also an $r$-powerful number.
\n If $r≥4$ then can the sum of $r-2$ coprime $r$-powerful numbers ever be itself $r$-powerful?
"},"Erdos939.erdos_939.variants.infinite":{"url":"/FormalConjectures/ErdosProblems/«939»/#Erdos939___erdos_939___variants___infinite","anchor":"Erdos939___erdos_939___variants___infinite","docHtml":"\n If $r≥4$ are there infinitely many sums of $r-2$ coprime $r$-powerful numbers\nthat are themselves $r$-powerful?
"},"Erdos939.erdos_939.variants.triples":{"url":"/FormalConjectures/ErdosProblems/«939»/#Erdos939___erdos_939___variants___triples","anchor":"Erdos939___erdos_939___variants___triples","docHtml":"\n Are there infinitely many triples of coprime $3$-powerful numbers $a, b, c$ such that $a + b = c$?
"},"Erdos939.erdos_939.variants.examples":{"url":"/FormalConjectures/ErdosProblems/«939»/#Erdos939___erdos_939___variants___examples","anchor":"Erdos939___erdos_939___variants___examples","docHtml":"\n Cambie has found several examples of the sum of $r - 2$ coprime $r$-powerful numbers being itself\n$r$-powerful. For example when $r=5$ we have\n$$3761^5=2^8\\cdot3^{10}\\cdot 5^7 + 2^{12}\\cdot 23^6 + 11^5\\cdot 13^5$$.
"},"Erdos939.erdos_939.variants.seven":{"url":"/FormalConjectures/ErdosProblems/«939»/#Erdos939___erdos_939___variants___seven","anchor":"Erdos939___erdos_939___variants___seven","docHtml":"\n Cambie has also found solutions when $r=7$.
"},"Erdos939.erdos_939.variants.eight":{"url":"/FormalConjectures/ErdosProblems/«939»/#Erdos939___erdos_939___variants___eight","anchor":"Erdos939___erdos_939___variants___eight","docHtml":"\n Cambie has also found solutions when $r=8$.
"},"Erdos939.erdos_939.variants.euler":{"url":"/FormalConjectures/ErdosProblems/«939»/#Erdos939___erdos_939___variants___euler","anchor":"Erdos939___erdos_939___variants___euler","docHtml":"\n Euler had conjectured that the sum of $k - 1$ many $k$-th powers is never a\n$k$-th power, but this is false for $k=5$, as Lander and Parkin [LaPa67] found\n$$27^5+84^5+110^5+133^5=144^5$$.
\n\n [LaPa67] Lander, L. J. and Parkin, T. R., \"A counterexample to Euler's sum of powers conjecture.\"\nMath. Comp. (1967), 101--103.
"},"Erdos325.IsSumThreePower":{"url":"/FormalConjectures/ErdosProblems/«325»/#Erdos325___IsSumThreePower","anchor":"Erdos325___IsSumThreePower","docHtml":"\n A predicate for $n$ to be the sum of three $k$th powers.
"},"Erdos325.cardIsSumThreePowerBelow":{"url":"/FormalConjectures/ErdosProblems/«325»/#Erdos325___cardIsSumThreePowerBelow","anchor":"Erdos325___cardIsSumThreePowerBelow","docHtml":"\n The number of integers $\\leq x$ which are the sum of three $k$th powers.
"},"Erdos325.erdos_325":{"url":"/FormalConjectures/ErdosProblems/«325»/#Erdos325___erdos_325","anchor":"Erdos325___erdos_325","docHtml":"\n Writing $f_{k, 3}(x)$ for the number of integers $\\leq x$ which are the sum of three $k$th powers,\nis it true that $f_{k, 3}(x) \\gg x ^ (3 / k)$?
"},"Erdos325.erdos_325.variants.weaker":{"url":"/FormalConjectures/ErdosProblems/«325»/#Erdos325___erdos_325___variants___weaker","anchor":"Erdos325___erdos_325___variants___weaker","docHtml":"\n Writing $f_{k, 3}(x)$ for the number of integers $\\leq x$ which are the sum of three $k$th powers,\nis it even true that $f_{k, 3}(x) \\gg_{\\epsilon} x ^ (3 / k - \\epsilon)$?
"},"Erdos325.erdos_325.variants.wooley":{"url":"/FormalConjectures/ErdosProblems/«325»/#Erdos325___erdos_325___variants___wooley","anchor":"Erdos325___erdos_325___variants___wooley","docHtml":"\n For $k = 3$, the best known is due to Wooley [Wo15]\n[Wo15] Wooley, Trevor D., Sums of three cubes, II. Acta Arith. (2015), 73-100.
"},"Erdos26.IsThick":{"url":"/FormalConjectures/ErdosProblems/«26»/#Erdos26___IsThick","anchor":"Erdos26___IsThick","docHtml":"\n A sequence of naturals $(a_i)$ is
\n The set of multiples of a sequence $(a_i)$ is ${na_i | n \\in \\mathbb{N}, i}$.
"},"Erdos26.multiplesOf_eq_univ":{"url":"/FormalConjectures/ErdosProblems/«26»/#Erdos26___multiplesOf_eq_univ","anchor":"Erdos26___multiplesOf_eq_univ"},"Erdos26.IsBehrend":{"url":"/FormalConjectures/ErdosProblems/«26»/#Erdos26___IsBehrend","anchor":"Erdos26___IsBehrend","docHtml":"\n A sequence of naturals $(a_i)$ is
\n A sequence of naturals $(a_i)$ is
\n Let $A\\subset\\mathbb{N}$ be infinite such that $\\sum_{a \\in A} \\frac{1}{a} = \\infty$. Must\nthere exist some $k\\geq 1$ such that almost all integers have a divisor of the form $a+k$\nfor some $a\\in A$?
\n\n This was formalized in Lean by Alexeev using Aristotle.
"},"Erdos26.erdos_26.variants.rusza":{"url":"/FormalConjectures/ErdosProblems/«26»/#Erdos26___erdos_26___variants___rusza","anchor":"Erdos26___erdos_26___variants___rusza","docHtml":"\n If we allow for $\\sum_{a\\in A} \\frac{1}{a} < \\infty$ then Rusza has found a counter-example.
"},"Erdos26.erdos_26.variants.tenenbaum":{"url":"/FormalConjectures/ErdosProblems/«26»/#Erdos26___erdos_26___variants___tenenbaum","anchor":"Erdos26___erdos_26___variants___tenenbaum","docHtml":"\n Tenenbaum asked the weaker variant where for every $\\epsilon>0$ there is\nsome $k=k(\\epsilon)$ such that at least $1-\\epsilon$ density of all integers have a\ndivisor of the form $a+k$ for some $a\\in A$.
\n\n The DeepMind prover agent has found a formal disproof of this statement.
"},"Erdos399.erdos_399":{"url":"/FormalConjectures/ErdosProblems/«399»/#Erdos399___erdos_399","anchor":"Erdos399___erdos_399","docHtml":"\n Is it true that there are no solutions to n! = x^k ± y^k with x,y,n ∈ ℕ, x*y > 1, and\nk > 2?
\n The answer is no: Jonas Barfield found the counterexample 10! = 48^4 - 36^4 (equivalently,\n10! + 36^4 = 48^4).
\n This is discussed in problem D2 of Guy's collection [Gu04].
\n\n This was formalized in Lean by Lu using Codex.
"},"Erdos399.erdos_399.variants.erdos_oblath":{"url":"/FormalConjectures/ErdosProblems/«399»/#Erdos399___erdos_399___variants___erdos_oblath","anchor":"Erdos399___erdos_399___variants___erdos_oblath","docHtml":"\n Erdős and Obláth [ErOb37] proved this is true when $(x,y)=1$ and $k\\neq 4$.
"},"Erdos399.erdos_399.variants.pollack_shapiro":{"url":"/FormalConjectures/ErdosProblems/«399»/#Erdos399___erdos_399___variants___pollack_shapiro","anchor":"Erdos399___erdos_399___variants___pollack_shapiro","docHtml":"\n Pollack and Shapiro [PoSh73] proved there are no solutions to $n!=x^4-1$.
"},"Erdos399.erdos_399.variants.cambie":{"url":"/FormalConjectures/ErdosProblems/«399»/#Erdos399___erdos_399___variants___cambie","anchor":"Erdos399___erdos_399___variants___cambie","docHtml":"\n Cambie has also observed that considerations modulo $8$ rule out any solutions to $n!=x^4+y^4$ with\n$(x,y)=1$ and $xy>1$.
"},"Erdos399.erdos_399.variants.sum_two_squares":{"url":"/FormalConjectures/ErdosProblems/«399»/#Erdos399___erdos_399___variants___sum_two_squares","anchor":"Erdos399___erdos_399___variants___sum_two_squares","docHtml":"\n Erdős and Obláth observed that the Bertrand-style fact (first proved by Breusch [Br32]) that, if\n$q_i$ is the sequence of primes congruent to $3\\pmod{4}$ then $q_{i+1}<2q_i$ except for $q_1=3$,\ntogether with Fermat's theorem on the sums of two squares implies that the only solution to\n$n!=x^2+y^2$ is $6!=12^2+24^2$.
"},"Erdos846.NonTrilinearFor":{"url":"/FormalConjectures/ErdosProblems/«846»/#Erdos846___NonTrilinearFor","anchor":"Erdos846___NonTrilinearFor","docHtml":"\n We say a subset A of points in the plane is ε-non-trilinear if any subset\nB of A, contains a non-trilinear subset C of size at least ε|B|.
\n We say a subset A of points in the plane is weakly non-trilinear if it is\na finite union of non-trilinear sets.
\nErdős Problem 846\nLet A ⊂ ℝ² be an infinite set for which there exists some ϵ>0 such that in any subset of A\nof size n there are always at least ϵn with no three on a line.\nIs it true that A is the union of a finite number of sets where no three are on a line?
\n In other words, prove or disprove the following statement: every infinite ε-non-trilinear subset of the\nplane is weakly non-trilinar.
\n Let $R(N)$ be the size of the largest $A\\subseteq{1, ..., N}$ such that all sums\n$\\sum_{n\\in S} \\frac{1}{n}$ are distinct for $S\\subseteq A$.
"},"Erdos321.erdos_321":{"url":"/FormalConjectures/ErdosProblems/«321»/#Erdos321___erdos_321","anchor":"Erdos321___erdos_321","docHtml":"\n Let $R(N)$ be the size of the largest $A\\subseteq{1, ..., N}$ such that all sums\n$\\sum_{n\\in S} \\frac{1}{n}$ are distinct for $S\\subseteq A$. What is $R(N)$?
"},"Erdos321.erdos_321.variants.isTheta":{"url":"/FormalConjectures/ErdosProblems/«321»/#Erdos321___erdos_321___variants___isTheta","anchor":"Erdos321___erdos_321___variants___isTheta","docHtml":"\n Let $R(N)$ be the size of the largest $A\\subseteq{1, ..., N}$ such that all sums\n$\\sum_{n\\in S} \\frac{1}{n}$ are distinct for $S\\subseteq A$. What is $\\Theta(R(N))$?
"},"Erdos321.erdos_321.variants.isBigO":{"url":"/FormalConjectures/ErdosProblems/«321»/#Erdos321___erdos_321___variants___isBigO","anchor":"Erdos321___erdos_321___variants___isBigO","docHtml":"\n Let $R(N)$ be the size of the largest $A\\subseteq{1, ..., N}$ such that all sums $\\sum_{n\\in S} \\frac{1}{n}$ are distinct for $S\\subseteq A$. Find the simplest $g(N)$ such that $R(N) = O(g(N))$.
"},"Erdos321.erdos_321.variants.isLittleO":{"url":"/FormalConjectures/ErdosProblems/«321»/#Erdos321___erdos_321___variants___isLittleO","anchor":"Erdos321___erdos_321___variants___isLittleO","docHtml":"\n Let $R(N)$ be the size of the largest $A\\subseteq{1, ..., N}$ such that all sums $\\sum_{n\\in S} \\frac{1}{n}$ are distinct for $S\\subseteq A$. Find the simplest $g(N)$ such that $R(N) = o(g(N))$.
"},"Erdos321.erdos_321.variants.lower":{"url":"/FormalConjectures/ErdosProblems/«321»/#Erdos321___erdos_321___variants___lower","anchor":"Erdos321___erdos_321___variants___lower","docHtml":"\n Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that\n$$\n\\frac{N}{\\log N} \\prod_{i=3}^{k} \\log_i N \\le R(N),\n$$\nvalid for any $k \\ge 4$ with $\\log_k N \\ge k$ and any $r \\ge 1$ with $\\log_{2r} N \\ge 1$. (In these bounds $\\log_i n$ denotes the $i$-fold iterated logarithm.)
\n\n [BlEr75] Bleicher, M. N. and Erdős, P.,
\n Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that\n$$\nR(N) \\le \\frac{1}{\\log 2} \\log_r N \\left( \\frac{N}{\\log N} \\prod_{i=3}^{r} \\log_i N \\right),\n$$\nvalid for any $k \\ge 4$ with $\\log_k N \\ge k$ and any $r \\ge 1$ with $\\log_{2r} N \\ge 1$. (In these bounds $\\log_i n$ denotes the $i$-fold iterated logarithm.)
\n\n [BlEr75] Bleicher, M. N. and Erdős, P.,
\n Let $\\frac a b\\in \\mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1 < n_1 < \\dots < n_k$,\neach the product of two distinct primes, such that $\\frac{a}{b}=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k}$?
"},"Erdos306.erdos_306.variants.integer_three_primes":{"url":"/FormalConjectures/ErdosProblems/«306»/#Erdos306___erdos_306___variants___integer_three_primes","anchor":"Erdos306___erdos_306___variants___integer_three_primes","docHtml":"\n Every positive integer can be expressed as an Egyptian fraction where each denominator is the\nproduct of three distinct primes.
"},"Erdos341.erdos_341":{"url":"/FormalConjectures/ErdosProblems/«341»/#Erdos341___erdos_341","anchor":"Erdos341___erdos_341","docHtml":"\n Let $A={a_1 < \\cdots < a_k}$ be a finite set of integers and extend it to an infinite\nsequence $\\overline{A}={a_1 < a_2 < \\cdots }$ by defining $a_{n+1}$ for $n \\geq k$ to be\nthe least integer exceeding $a_n$ which is not of the form $a_i + a_j$ with $i,j \\leq n$.\nIs it true that the sequence of differences $a_{m+1}-a_m$ is eventually periodic?
\n\n This problem is discussed under Problem 7 on Green's open problems list.
"},"Erdos52.erdos_52":{"url":"/FormalConjectures/ErdosProblems/«52»/#Erdos52___erdos_52","anchor":"Erdos52___erdos_52","docHtml":"\n Let $A$ be a finite set of integers. Is it true that for every $\\epsilon>0$\n$\\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg_\\epsilon \\lvert A\\rvert^{2-\\epsilon}?$
"},"Erdos602.IsMonochromatic":{"url":"/FormalConjectures/ErdosProblems/«602»/#Erdos602___IsMonochromatic","anchor":"Erdos602___IsMonochromatic","docHtml":"\n A set A ⊆ α is monochromatic under a 2-colouring f : α → Fin 2\nif all elements of A receive the same colour.
\n A family (A_i)_{i ∈ I} of subsets of α has Property B if there exists\na 2-colouring f : α → Fin 2 such that no A_i is monochromatic.
\n Does every almost-disjoint family of countably infinite sets whose pairwise\nintersections all have size ≠ 1 have Property B?
\n\n Formally: let α be any type, let (A_i)_{i ∈ I} be a family of countably infinite subsets\nof α such that for all i ≠ j, the intersection A_i ∩ A_j is finite and\n|A_i ∩ A_j| ≠ 1. Does there exist a 2-colouring f : α → Fin 2 such that no A_i is\nmonochromatic?
\n This is an open question about Property B for almost-disjoint families with a\nforbidden intersection size of 1.
\n\nNote: This generalises the formulation in which the ground set is ℕ. Since every\ncountably infinite set is in bijection with ℕ, the two formulations are equivalent, but\nworking over an arbitrary ground type makes the statement apply immediately to, e.g.,\nalmost-disjoint families of countable subsets of an uncountable space.
\nTrivial case: pairwise disjoint families.
\n\n If the A_i are pairwise disjoint (all intersections are empty, which in\nparticular satisfies |A_i ∩ A_j| ≠ 1), then Property B holds trivially.
\nProof sketch: Since each A_i is infinite, it has (at least) two distinct elements\na_i and b_i. We can define a colouring that assigns colour 0 to a_i and colour 1\nto b_i for each i (using disjointness, these choices don't conflict), and extend\narbitrarily elsewhere. Then no A_i is monochromatic.
\nCountable index set case.
\n\n If the index set is countable, the answer is yes, and the intersection\ncondition is unnecessary. This is Bernstein's Lemma:\nevery countable system of infinite sets has Property B.
"},"Erdos602.erdos_602.variants.single_set":{"url":"/FormalConjectures/ErdosProblems/«602»/#Erdos602___erdos_602___variants___single_set","anchor":"Erdos602___erdos_602___variants___single_set","docHtml":"\nIntersections of size ≥ 2 suffice.
\n\n For a single countably infinite set A ⊆ α, there trivially exists a 2-colouring\nof α that makes A non-monochromatic: since A is infinite, it has two distinct\nelements, so any colouring that assigns them different colours works.
\nEmpty index set.
\n\n If the index set I is empty (has no elements), then Property B holds vacuously:\nany 2-colouring works, since there are no sets to be made non-monochromatic.
\nUnique index set.
\n\n If the index set has exactly one element (i.e., [Unique I]), then Property B holds:\nany 2-colouring that makes the single set A (default : I) non-monochromatic works.\nThis follows from the single-set case.
\nTwo infinite sets with pairwise intersection of size ≠ 1.
\n\n If the family consists of exactly two countably infinite sets A₀ and A₁ with\n|A₀ ∩ A₁| ≠ 1 (and finite), then Property B holds.
\nProof sketch:
\n\n If A₀ ∩ A₁ = ∅: the sets are disjoint. Pick distinct a, b ∈ A₀ and distinct\nc, d ∈ A₁. Colour b and c with 1, everything else with 0. Then A₀ has\na (colour 0) and b (colour 1), and A₁ has c (colour 1) and d (colour 0),\nso neither is monochromatic.
\n If |A₀ ∩ A₁| ≥ 2: the intersection contains two distinct points x and y.\nAssign x colour 0 and y colour 1. Both A₀ and A₁ contain x and y,\nso neither is monochromatic.
\n A natural but FALSE relaxation of erdos_602.variants.disjoint: drop the\nhypothesis that each A i is infinite. The original disjoint variant requires\n(∀ i, (A i).Infinite). Without it, the claim is false.
\n Formal disproof of disjoint_without_infinite_claim.
\nCounterexample: Take α = ℕ, I = Fin 2, with A 0 = {0} and A 1 = {1}.\nThese are pairwise disjoint, satisfying the only hypothesis. But singleton sets\nare vacuously monochromatic under any colouring: the only pair (x, y) ∈ {0} × {0}\nis (0, 0), and f 0 = f 0 trivially. So any colouring makes A 0 monochromatic,\nmeaning HasPropertyB fails.
\n Let $f(n,k)$ be minimal such that every $F$ family of $n$-uniform sets with $|F| \\ge f(n,k)$\ncontains a $k$-sunflower.
"},"Erdos20.f_0_1":{"url":"/FormalConjectures/ErdosProblems/«20»/#Erdos20___f_0_1","anchor":"Erdos20___f_0_1"},"Erdos20.erdos_20":{"url":"/FormalConjectures/ErdosProblems/«20»/#Erdos20___erdos_20","anchor":"Erdos20___erdos_20","docHtml":"\n Is it true that $f(n,k) < c_k^n$ for some constant $c_k>0$ and for all $n > 0$?
"},"Erdos647.erdos_647":{"url":"/FormalConjectures/ErdosProblems/«647»/#Erdos647___erdos_647","anchor":"Erdos647___erdos_647","docHtml":"\n Let $\\tau(n)$ count the number of divisors of $n$. Is there some $n > 24$ such that\n$$\n\\max_{m < n}(m + \\tau(m)) \\leq n + 2?\n$$
"},"Erdos647.erdos_647.variants.twenty_four":{"url":"/FormalConjectures/ErdosProblems/«647»/#Erdos647___erdos_647___variants___twenty_four","anchor":"Erdos647___erdos_647___variants___twenty_four","docHtml":"\n This is true for $n = 24$.
"},"Erdos647.erdos_647.variants.lim":{"url":"/FormalConjectures/ErdosProblems/«647»/#Erdos647___erdos_647___variants___lim","anchor":"Erdos647___erdos_647___variants___lim","docHtml":"\n Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in\nfact suggests that\n$$\nlim_{n\\to\\infty} \\max_{m < n}(\\tau(m) + m − n) = \\infty.\n$$
"},"Erdos647.erdos_647.variants.infinite":{"url":"/FormalConjectures/ErdosProblems/«647»/#Erdos647___erdos_647___variants___infinite","anchor":"Erdos647___erdos_647___variants___infinite","docHtml":"\n Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$\nfor which\n$$\n\\max_{n−k < m < n}(m + \\tau(m)) ≤ n + 2.\n$$
"},"Erdos659.erdos_659":{"url":"/FormalConjectures/ErdosProblems/«659»/#Erdos659___erdos_659","anchor":"Erdos659___erdos_659","docHtml":"\n Is there a set of $n$ points in $\\mathbb{R}^2$ such that every subset of $4$ points determines at\nleast $3$ distances, yet the total number of distinct distances is $\\ll \\frac{n}{\\sqrt{\\log n}}$?
\n\n There does exist such a set: a suitable truncation of the lattice\n${(a,b\\sqrt{2}): a,b\\in\\mathbb{Z}}$ suffices. This construction appears to have been first\nconsidered by Moree and Osburn \\cite{MoOs06}, who proved that it has\n$\\ll \\frac{n}{\\sqrt{\\log n}}$ many distinct distances. This construction was independently found by\nLund and Sheffer,\nwho further noted that this configuration contains no squares or equilateral triangles.
\n\n There are only six possible configurations of $4$ points which determine only $2$ distances\n(first noted by Erdős and Fishburn [ErFi96]), and five of them contain either a square or an\nequilateral triangle. The remaining configuration contains four points from a regular pentagon,\nand Grayzel [Gr26] (using Gemini) has noted in the comments that this configuration can also be\nruled out, thus giving a complete solution to this problem.
\n\n Boris Alexeev provides a formalisation of the reduction, which is conditional on Bernays' theorem\n(assumed as an axiom in the proof to obtain the $O(n/\\sqrt{\\log n})$ bound).\nSee the formal proof.
"},"Erdos1095.g":{"url":"/FormalConjectures/ErdosProblems/«1095»/#Erdos1095___g","anchor":"Erdos1095___g","docHtml":"\n Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\\binom{n}{k}$ are $>k$.
"},"Erdos1095.erdos_1095.variants.lower_solved":{"url":"/FormalConjectures/ErdosProblems/«1095»/#Erdos1095___erdos_1095___variants___lower_solved","anchor":"Erdos1095___erdos_1095___variants___lower_solved","docHtml":"\n The current record is $g(k) \\gg \\exp(c(\\log k)^2)$ for some $c>0$, due to Konyagin [Ko99b]. -
"},"Erdos1095.erdos_1095.variants.upper_conjecture":{"url":"/FormalConjectures/ErdosProblems/«1095»/#Erdos1095___erdos_1095___variants___upper_conjecture","anchor":"Erdos1095___erdos_1095___variants___upper_conjecture","docHtml":"\n Ecklund, Erdős, and Selfridge [EES74] conjectured $g(k)\\leq \\exp((1+o(1))k)$.
"},"Erdos1095.erdos_1095.variants.lower_conjecture":{"url":"/FormalConjectures/ErdosProblems/«1095»/#Erdos1095___erdos_1095___variants___lower_conjecture","anchor":"Erdos1095___erdos_1095___variants___lower_conjecture","docHtml":"\n Erdős, Lacampagne, and Selfridge [ELS93] write 'it is clear to every right-thinking person' that\n$g(k)\\geq\\exp(c\\frac{k}{\\log k})$ for some constant $c>0$.
"},"Erdos1095.erdos_1095.variants.log_equivalent":{"url":"/FormalConjectures/ErdosProblems/«1095»/#Erdos1095___erdos_1095___variants___log_equivalent","anchor":"Erdos1095___erdos_1095___variants___log_equivalent","docHtml":"\n Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that $\\log g(k) \\asymp \\frac{k}{\\log k}$.
"},"Erdos1137.erdos_1137":{"url":"/FormalConjectures/ErdosProblems/«1137»/#Erdos1137___erdos_1137","anchor":"Erdos1137___erdos_1137","docHtml":"\n Let $d_n=p_{n+1}-p_n$, where $p_n$ denotes the $n$th prime. Is it true that\n$$\\frac{\\max_{n < x}d_{n}d_{n-1}}{(\\max_{n < x}d_n)^2}\\to 0$$ as $x\\to \\infty$?
"},"Erdos329.sqrtPartialDensity":{"url":"/FormalConjectures/ErdosProblems/«329»/#Erdos329___sqrtPartialDensity","anchor":"Erdos329___sqrtPartialDensity","docHtml":"\n The partial density of a Sidon set A up to N, normalized by dividing by √N instead of N.\nThis measures how close the set comes to the optimal density for Sidon sets.
\n The upper density of a Sidon set A, normalized by √N.
\nErdős Problem 329.\nLet A ⊆ ℕ be a Sidon set. How large can\nlim sup_{N → ∞} |A ∩ {1,…,N}| / N^{1/2}\nbe?
\n Erdős proved that upper density 1 / 2 can be attained; in particular,\nthere exists a Sidon set whose upper density is 1 / 2.
\n Krückeberg ([Kr61]) exhibited an infinite Sidon set A with\nsidonUpperDensity A = 1 / Real.sqrt 2, improving Erdős’ earlier\n1 / 2 lower bound.
\n [Kr61] Krückeberg, Fritz, $B\\sb{2}$-Folgen und verwandte Zahlenfolgen. J. Reine Angew. Math. (1961), 53-60.
"},"Erdos329.erdos_329.variants.turan_1941":{"url":"/FormalConjectures/ErdosProblems/«329»/#Erdos329___erdos_329___variants___turan_1941","anchor":"Erdos329___erdos_329___variants___turan_1941","docHtml":"\n Erdős and Turán [ErTu41] proved the upper bound of 1.
\n\n [ErTu41] Erdős, P. and Turán, P., On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc. (1941), 212-215.
"},"Erdos329.erdos_329.variants.of_sub_perfectDifferenceSet":{"url":"/FormalConjectures/ErdosProblems/«329»/#Erdos329___erdos_329___variants___of_sub_perfectDifferenceSet","anchor":"Erdos329___erdos_329___variants___of_sub_perfectDifferenceSet","docHtml":"\n If any finite Sidon set can be embedded in a perfect difference set,\nthen the maximum density would be 1.
"},"Erdos329.erdos_329.variants.converse_implication":{"url":"/FormalConjectures/ErdosProblems/«329»/#Erdos329___erdos_329___variants___converse_implication","anchor":"Erdos329___erdos_329___variants___converse_implication","docHtml":"\n The converse: if the maximum density is 1, then any finite Sidon set\ncan be embedded in a perfect difference set.
"},"Erdos329.exists_sidon_pos_density":{"url":"/FormalConjectures/ErdosProblems/«329»/#Erdos329___exists_sidon_pos_density","anchor":"Erdos329___exists_sidon_pos_density","docHtml":"\n It is possible to construct a Sidon set with positive density.
"},"Erdos463.erdos_463":{"url":"/FormalConjectures/ErdosProblems/«463»/#Erdos463___erdos_463","anchor":"Erdos463___erdos_463","docHtml":"\n Is there a function $f$ with $f(n)\\to\\infty$ as $n\\to\\infty$ such that,\nfor all large $n$, there is a composite number $m$ such that\n$$\nn + f(n) < m < n + p(m)\n$$\nHere $p(m)$ is the least prime factor of $m$.
"},"Erdos295.exists_k":{"url":"/FormalConjectures/ErdosProblems/«295»/#Erdos295___exists_k","anchor":"Erdos295___exists_k","docHtml":"\n Helper lemma: for each $N$, there exists $k$ and $n_1 < ... < n_k$ such that\n$N ≤ n_1 < ⋯ < n_k$ with $\\frac 1 {n_1} + ... + \\frac 1 {n_k} = 1$.
"},"Erdos295.k":{"url":"/FormalConjectures/ErdosProblems/«295»/#Erdos295___k","anchor":"Erdos295___k","docHtml":"\n Let $k(N)$ denote the smallest $k$ such that there exists\n$N ≤ n_1 < ⋯ < n_k$ with $\\frac 1 {n_1} + ... + \\frac 1 {n_k} = 1$.
"},"Erdos295.erdos_295":{"url":"/FormalConjectures/ErdosProblems/«295»/#Erdos295___erdos_295","anchor":"Erdos295___erdos_295","docHtml":"\n Let $k(N)$ denote the smallest $k$ such that there exists\n$N ≤ n_1 < ⋯ < n_k$ with $\\frac 1 {n_1} + ... + \\frac 1 {n_k} = 1$
\n\n Is it true that $\\lim_{N \\to \\infty} k(N) - (e - 1)N = \\infty$?
"},"Erdos295.erdos_295.variants.erdos_straus":{"url":"/FormalConjectures/ErdosProblems/«295»/#Erdos295___erdos_295___variants___erdos_straus","anchor":"Erdos295___erdos_295___variants___erdos_straus","docHtml":"\n Erdős and Straus have proved the existence of some constant $c>0$\nsuch that $-c < k(N)-(e-1)N \\ll \\frac N {\\log N}$
"},"Erdos303.erdos_303":{"url":"/FormalConjectures/ErdosProblems/«303»/#Erdos303___erdos_303","anchor":"Erdos303___erdos_303","docHtml":"\n Is it true that in any finite colouring of the integers there exists a monochromatic solution\nto $\\frac 1 a = \\frac 1 b + \\frac 1 c$ with distinct $a, b, c$?
\n\n This is true, as proved by Brown and Rödl [BrRo91].
\n\n This was formalized in Lean by Yuan using Seed-Prover.
"},"Erdos142.r":{"url":"/FormalConjectures/ErdosProblems/«142»/#Erdos142___r","anchor":"Erdos142___r","docHtml":"\n
\n Prove an asymptotic formula for $r_k(N)$, the largest possible size of a subset\nof ${1, \\dots, N}$ that does not contain any non-trivial $k$-term arithmetic progression.
"},"Erdos142.erdos_142.variants.lower":{"url":"/FormalConjectures/ErdosProblems/«142»/#Erdos142___erdos_142___variants___lower","anchor":"Erdos142___erdos_142___variants___lower","docHtml":"\n Show that $r_k(N) = o_k(N / \\log N)$, where $r_k(N)$ the largest possible size of a subset\nof ${1, \\dots, N}$ that does not contain any non-trivial $k$-term arithmetic progression.
"},"Erdos142.erdos_142.variants.upper":{"url":"/FormalConjectures/ErdosProblems/«142»/#Erdos142___erdos_142___variants___upper","anchor":"Erdos142___erdos_142___variants___upper","docHtml":"\n Find functions $f_k$, such that $r_k(N) = O_k(f_k)$, where $r_k(N)$ the largest possible size of a\nsubset of ${1, \\dots, N}$ that does not contain any non-trivial $k$-term arithmetic progression.
"},"Erdos142.erdos_142.variants.three":{"url":"/FormalConjectures/ErdosProblems/«142»/#Erdos142___erdos_142___variants___three","anchor":"Erdos142___erdos_142___variants___three","docHtml":"\n Prove an asymptotic formula for $r_3(N)$, the largest possible size of a subset\nof ${1, \\dots, N}$ that does not contain any non-trivial $3$-term arithmetic progression.
"},"Erdos1096.erdos_1096":{"url":"/FormalConjectures/ErdosProblems/«1096»/#Erdos1096___erdos_1096","anchor":"Erdos1096___erdos_1096","docHtml":"\n Let $1<q<1+\\epsilon$ and consider the set of numbers of the shape\n$\\sum_{i\\in S}q^i$ (for all finite $S$), ordered by size as\n$0=x_1<x_2<\\cdots$.
\n\n Is it true that, provided $\\epsilon>0$ is sufficiently small, $x_{k+1}-x_k \\to 0$?
\n\n This was solved affirmatively by Erdős and Komornik [ErKo98], who proved the conclusion\nwhenever $1<q<\\sqrt{q_1}$, where $q_1$ is the second Pisot-Vijayaraghavan number.
"},"Erdos932.erdos_932":{"url":"/FormalConjectures/ErdosProblems/«932»/#Erdos932___erdos_932","anchor":"Erdos932___erdos_932","docHtml":"\n Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two\nintegers $p_r < n < p_{r+1}$ all of whose prime factors are $< p_{r + 1} - p_r$.
"},"Erdos932.erdos_932.variants.one_le":{"url":"/FormalConjectures/ErdosProblems/«932»/#Erdos932___erdos_932___variants___one_le","anchor":"Erdos932___erdos_932___variants___one_le","docHtml":"\n Erdős could show that the density of $r$ such that at least one such $n$ exists is $0$.
"},"Erdos317.erdos_317":{"url":"/FormalConjectures/ErdosProblems/«317»/#Erdos317___erdos_317","anchor":"Erdos317___erdos_317","docHtml":"\n Is there some constant $c>0$ such that for every $n\\geq 1$ there exists some $\\delta_k\\in {-1,0,1}$ for $1\\leq k\\leq n$ with\n$$0< \\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert < \\frac{c}{2^n}?$$
"},"Erdos317.erdos_317.variants.claim2":{"url":"/FormalConjectures/ErdosProblems/«317»/#Erdos317___erdos_317___variants___claim2","anchor":"Erdos317___erdos_317___variants___claim2","docHtml":"\n Is it true that for sufficiently large $n$, for any $\\delta_k\\in {-1,0,1}$,\n$$\\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert > \\frac{1}{[1,\\ldots,n]}$$\nwhenever the left-hand side is not zero?
"},"Erdos317.claim2_inequality":{"url":"/FormalConjectures/ErdosProblems/«317»/#Erdos317___claim2_inequality","anchor":"Erdos317___claim2_inequality","docHtml":"\n Inequality in erdos_317.variants.claim2 is obvious, the problem is strict inequality.
\nerdos_317.variants.claim2 fails for small $n$, for example\n$$\\frac{1}{2}-\\frac{1}{3}-\\frac{1}{4}=-\\frac{1}{12}.$$
\n A finite set of naturals $A$ is said to be a sum-distinct set for $N \\in \\mathbb{N}$ if\n$A\\subseteq{1, ..., N}$ and the sums $\\sum_{a\\in S}a$ are distinct for all $S\\subseteq A$
"},"Erdos1.erdos_1":{"url":"/FormalConjectures/ErdosProblems/«1»/#Erdos1___erdos_1","anchor":"Erdos1___erdos_1","docHtml":"\n If $A\\subseteq{1, ..., N}$ with $|A| = n$ is such that the subset sums $\\sum_{a\\in S}a$ are\ndistinct for all $S\\subseteq A$ then\n$$\nN \\gg 2 ^ n.\n$$
"},"Erdos1.erdos_1.variants.weaker":{"url":"/FormalConjectures/ErdosProblems/«1»/#Erdos1___erdos_1___variants___weaker","anchor":"Erdos1___erdos_1___variants___weaker","docHtml":"\n The trivial lower bound is $N \\gg 2^n / n$.
"},"Erdos1.erdos_1.variants.lb":{"url":"/FormalConjectures/ErdosProblems/«1»/#Erdos1___erdos_1___variants___lb","anchor":"Erdos1___erdos_1___variants___lb","docHtml":"\n Erdős and Moser [Er56] proved\n$$\nN \\geq (\\tfrac{1}{4} - o(1)) \\frac{2^n}{\\sqrt{n}}.\n$$
\n\n [Er56] Erdős, P.,
\n A number of improvements of the constant $\\frac{1}{4}$ have been given, with the current\nrecord $\\sqrt{2 / \\pi}$ first provied in unpublished work of Elkies and Gleason.
"},"Erdos1.IsSumDistinctRealSet":{"url":"/FormalConjectures/ErdosProblems/«1»/#Erdos1___IsSumDistinctRealSet","anchor":"Erdos1___IsSumDistinctRealSet","docHtml":"\n A finite set of real numbers is said to be sum-distinct if all the subset sums differ by\nat least $1$.
"},"Erdos1.erdos_1.variants.real":{"url":"/FormalConjectures/ErdosProblems/«1»/#Erdos1___erdos_1___variants___real","anchor":"Erdos1___erdos_1___variants___real","docHtml":"\n A generalisation of the problem to sets $A \\subseteq (0, N]$ of real numbers, such that the subset\nsums all differ by at least $1$ is proposed in [Er73] and [ErGr80].
\n\n [Er73] Erdős, P.,
\n [ErGr80] Erdős, P. and Graham, R.,
\n The minimal value of $N$ such that there exists a sum-distinct set with three\nelements is $4$.
\n\n https://oeis.org/A276661
"},"Erdos1.erdos_1.variants.least_N_5":{"url":"/FormalConjectures/ErdosProblems/«1»/#Erdos1___erdos_1___variants___least_N_5","anchor":"Erdos1___erdos_1___variants___least_N_5","docHtml":"\n The minimal value of $N$ such that there exists a sum-distinct set with five\nelements is $13$.
\n\n https://oeis.org/A276661
"},"Erdos1.erdos_1.variants.least_N_9":{"url":"/FormalConjectures/ErdosProblems/«1»/#Erdos1___erdos_1___variants___least_N_9","anchor":"Erdos1___erdos_1___variants___least_N_9","docHtml":"\n The minimal value of $N$ such that there exists a sum-distinct set with nine\nelements is $161$.
\n\n https://oeis.org/A276661
"},"Erdos229.erdos_229":{"url":"/FormalConjectures/ErdosProblems/«229»/#Erdos229___erdos_229","anchor":"Erdos229___erdos_229","docHtml":"\n Let $(S_n)_{n \\ge 1}$ be a sequence of sets of complex numbers, none of which have a finite\nlimit point. Does there exist an entire transcendental function $f(z)$ such that, for all $n \\ge 1$, there\nexists some $k_n \\ge 0$ such that $f^{(k_n)}(z) = 0$ for all $z \\in S_n$.
\n\n This is Problem 2.30 in [Ha74], where it is attributed to Erdős.
\n\n Solved in the affirmative by Barth and Schneider [BaSc72].
\n\n This was formalized in Lean by Alexeev using Aristotle.
"},"Erdos229.theorem_1":{"url":"/FormalConjectures/ErdosProblems/«229»/#Erdos229___theorem_1","anchor":"Erdos229___theorem_1","docHtml":"\n Let ${S_k}$ be any sequence of sets in the complex plane, each of which has no finite\nlimit point. Then there exists a sequence ${n_k}$ of positive integers and a\ntranscendental entire function $f(z)$ such that $f^{(n_k)}(z) = 0$ if $z \\in S_k$.
"},"Erdos591.erdos_591":{"url":"/FormalConjectures/ErdosProblems/«591»/#Erdos591___erdos_591","anchor":"Erdos591___erdos_591","docHtml":"\n Let $α$ be the infinite ordinal $\\omega^{\\omega^2}$. Is it true that any red/blue colouring of the\nedges of $K_α$ there is either a red $K_α$ or a blue $K_3$?
\n\n This is true and was proved independently by Schipperus [Sc10] and Darby.
"},"Erdos82.IsRegularInduced":{"url":"/FormalConjectures/ErdosProblems/«82»/#Erdos82___IsRegularInduced","anchor":"Erdos82___IsRegularInduced","docHtml":"\n A predicate that holds if $S$ is a regular induced subgraph of $G$
"},"Erdos82.F":{"url":"/FormalConjectures/ErdosProblems/«82»/#Erdos82___F","anchor":"Erdos82___F","docHtml":"\n $F(n)$ is the maximal integer such that every graph on $n$ vertices\ncontains a regular induced subgraph on at least $F(n)$ vertices.
"},"Erdos82.erdos_82":{"url":"/FormalConjectures/ErdosProblems/«82»/#Erdos82___erdos_82","anchor":"Erdos82___erdos_82","docHtml":"\n $F(n) / \\log n \\to \\infty as n \\to \\infty$
"},"Erdos82.erdos_82.variants.F_upper_bound":{"url":"/FormalConjectures/ErdosProblems/«82»/#Erdos82___erdos_82___variants___F_upper_bound","anchor":"Erdos82___erdos_82___variants___F_upper_bound","docHtml":"\n $F(n) \\le O(n^{1/2} \\ln ^ {3/4} n)$
\n\n Theorem 1.4 from [AKS07]
\n\n [AKS07] Alon, N. and Krivelevich, M. and Sudakov, B., Large nearly regular induced subgraphs. arXiv:0710.2106 (2007).
"},"Erdos848.NonSquarefreeProductProp":{"url":"/FormalConjectures/ErdosProblems/«848»/#Erdos848___NonSquarefreeProductProp","anchor":"Erdos848___NonSquarefreeProductProp","docHtml":"\n A set $A$ has the non-squarefree product property if $ab + 1$ is not squarefree\nfor all $a, b ∈ A$.
"},"Erdos848.A₇":{"url":"/FormalConjectures/ErdosProblems/«848»/#Erdos848___A___","anchor":"Erdos848___A___","docHtml":"\n The candidate extremal set: ${n ∈ {0, \\dots, N-1} : n ≡ 7 (mod 25)}$.
"},"Erdos848.Erdos848For":{"url":"/FormalConjectures/ErdosProblems/«848»/#Erdos848___Erdos848For","anchor":"Erdos848___Erdos848For","docHtml":"\n The Erdős Problem 848 statement for a fixed $N$: any set $A ⊆ {0, \\dots, N-1}$ with\nthe non-squarefree product property has cardinality at most $|A₇(N)|$.
"},"Erdos848.erdos_848":{"url":"/FormalConjectures/ErdosProblems/«848»/#Erdos848___erdos_848","anchor":"Erdos848___erdos_848","docHtml":"\n Is the maximum size of a set $A ⊆ {1, \\dots, N}$ such that $ab + 1$ is never squarefree\n(for all $a, b ∈ A$) achieved by taking those $n ≡ 7 \\pmod{25}$?
\n\n This asks whether Erdos848 N holds for all $N$ (formulated using A ⊆ Finset.range N).
\n This was solved for all sufficiently large $N$ by Sawhney in this note. In fact, Sawhney proves\nsomething slightly stronger, that there exists some constant $c>0$ such that if\n$\\lvert A\\rvert \\geq (\\frac{1}{25}-c)N$ and $N$ is large then $A$ is contained in either\n${ n\\equiv 7\\pmod{25}}$ or ${n\\equiv 18\\pmod{25}}$.
"},"Erdos848.erdos_848.variants.asymptotic":{"url":"/FormalConjectures/ErdosProblems/«848»/#Erdos848___erdos_848___variants___asymptotic","anchor":"Erdos848___erdos_848___variants___asymptotic","docHtml":"\n There exists $N₀$ such that for all $N ≥ N₀$, if $A ⊆ {1, \\dots, N}$ satisfies that $ab + 1$\nis never squarefree for all $a, b ∈ A$, then $|A| ≤ |{n ≤ N : n ≡ 7 \\pmod{25}}|$.
\n\n More precisely, Sawhney proves: there exist absolute constants $η > 0$ and $N₀$\nsuch that for all $N ≥ N₀$, if $|A| ≥ (1/25 - η)N$ then $A ⊆ {n : n ≡ 7 \\pmod{25}}$ or\n$A ⊆ {n : n ≡ 18 \\pmod{25}}$.
\n\n A complete formal Lean 4 proof is available at:\nhttps://github.com/The-Obstacle-Is-The-Way/erdos-banger
"},"Erdos349.IsGoodPair":{"url":"/FormalConjectures/ErdosProblems/«349»/#Erdos349___IsGoodPair","anchor":"Erdos349___IsGoodPair","docHtml":"\n This defines the core property of the problem: For what values of $t,\\alpha \\in (0,\\infty)$\nis the sequence $\\lfloor t\\alpha^n\\rfloor$ complete?
"},"Erdos349.erdos_349":{"url":"/FormalConjectures/ErdosProblems/«349»/#Erdos349___erdos_349","anchor":"Erdos349___erdos_349","docHtml":"\n For what values of $t,\\alpha \\in (0,\\infty)$ is the sequence $\\lfloor t\\alpha^n\\rfloor$ complete\n(that is, all sufficiently large integers are the sum of distinct integers of the form $\\lfloor t\\alpha^n\\rfloor$)?
"},"Erdos349.complete_for_alpha_in_Ioo_one_to_goldenRatio":{"url":"/FormalConjectures/ErdosProblems/«349»/#Erdos349___complete_for_alpha_in_Ioo_one_to_goldenRatio","anchor":"Erdos349___complete_for_alpha_in_Ioo_one_to_goldenRatio","docHtml":"\n It seems likely that the sequence is complete for all\nfor all $t>0$ and all $1 < \\alpha < \\frac{1+\\sqrt{5}}{2}$.
"},"Erdos349.exists_t_for_k_disjoint_segments":{"url":"/FormalConjectures/ErdosProblems/«349»/#Erdos349___exists_t_for_k_disjoint_segments","anchor":"Erdos349___exists_t_for_k_disjoint_segments","docHtml":"\n For any $k$ there exists some $t_k\\in (0,1)$ such that the set of $\\alpha$\nsuch that the sequence $\\lfloor t_k\\alpha^n\\rfloor$ is complete consists of at least $k$\ndisjoint line segments.
"},"Erdos349.erdos_349.variants.floor_3_halves_odd":{"url":"/FormalConjectures/ErdosProblems/«349»/#Erdos349___erdos_349___variants___floor_3_halves_odd","anchor":"Erdos349___erdos_349___variants___floor_3_halves_odd","docHtml":"\n Is it true that the terms of the sequence $\\lfloor (3/2)^n\\rfloor$ are odd infinitely\noften and even infinitely often?
"},"Erdos349.erdos_349.variants.floor_3_halves_even":{"url":"/FormalConjectures/ErdosProblems/«349»/#Erdos349___erdos_349___variants___floor_3_halves_even","anchor":"Erdos349___erdos_349___variants___floor_3_halves_even","docHtml":"\n Is it true that the terms of the sequence $\\lfloor (3/2)^n\\rfloor$ are even infinitely often?
"},"Erdos1052.properUnitaryDivisors":{"url":"/FormalConjectures/ErdosProblems/«1052»/#Erdos1052___properUnitaryDivisors","anchor":"Erdos1052___properUnitaryDivisors","docHtml":"\n A proper unitary divisor of $n$ is a divisor $d$ of $n$\nsuch that $d$ is coprime to $n/d$, and $d < n$.
"},"Erdos1052.IsUnitaryPerfect":{"url":"/FormalConjectures/ErdosProblems/«1052»/#Erdos1052___IsUnitaryPerfect","anchor":"Erdos1052___IsUnitaryPerfect","docHtml":"\n A number $n > 0$ is a unitary perfect number if it is the sum of its proper unitary divisors.
"},"Erdos1052.erdos_1052":{"url":"/FormalConjectures/ErdosProblems/«1052»/#Erdos1052___erdos_1052","anchor":"Erdos1052___erdos_1052","docHtml":"\n Are there only finitely many unitary perfect numbers?
"},"Erdos1052.even_of_isUnitaryPerfect":{"url":"/FormalConjectures/ErdosProblems/«1052»/#Erdos1052___even_of_isUnitaryPerfect","anchor":"Erdos1052___even_of_isUnitaryPerfect","docHtml":"\n All unitary perfect numbers are even.
\n\n Formal proof linked here provided by AlphaProof.
"},"Erdos1052.isUnitaryPerfect_6":{"url":"/FormalConjectures/ErdosProblems/«1052»/#Erdos1052___isUnitaryPerfect_6","anchor":"Erdos1052___isUnitaryPerfect_6"},"Erdos1052.isUnitaryPerfect_60":{"url":"/FormalConjectures/ErdosProblems/«1052»/#Erdos1052___isUnitaryPerfect_60","anchor":"Erdos1052___isUnitaryPerfect_60"},"Erdos1052.isUnitaryPerfect_90":{"url":"/FormalConjectures/ErdosProblems/«1052»/#Erdos1052___isUnitaryPerfect_90","anchor":"Erdos1052___isUnitaryPerfect_90"},"Erdos1052.isUnitaryPerfect_87360":{"url":"/FormalConjectures/ErdosProblems/«1052»/#Erdos1052___isUnitaryPerfect_87360","anchor":"Erdos1052___isUnitaryPerfect_87360"},"Erdos1052.isUnitaryPerfect_146361946186458562560000":{"url":"/FormalConjectures/ErdosProblems/«1052»/#Erdos1052___isUnitaryPerfect_146361946186458562560000","anchor":"Erdos1052___isUnitaryPerfect_146361946186458562560000"},"Erdos158.B2":{"url":"/FormalConjectures/ErdosProblems/«158»/#Erdos158___B2","anchor":"Erdos158___B2","docHtml":"\n A set A ⊆ ℕ is said to be a B₂[g] set if for all n, the equation\na + a' = n, a ≤ a', a, a' ∈ A has at most g solutions. This is defined in [ESS94].
\n A set is B₂[1] iff it is Sidon.
\n Let A be an infinite B₂[2] set. Must liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0?
\n Let A be an infinite Sidon set. Then\nliminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) * (log N) ^ (1 / 2) < ∞. This is proved in [ESS94].
\n As a corollary of erdos_158.isSidon', we can prove that\nliminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0 for any infinite Sidon set A.
\n Does there exists an entire non-zero transcendental function f : ℂ → ℂ such that for any\nsequence n₀ < n₁ < ..., { z | ∃ k, iteratedDeriv (n k) f z = 0 } is dense.
\n Is there and $A \\subset \\mathbb{N}$ is such that\n$$\\lim_{n\\to \\infty}\\frac{1_A\\ast 1_A(n)}{\\log n}$$\nexists and is $\\ne 0$?
"},"Erdos318.P₁":{"url":"/FormalConjectures/ErdosProblems/«318»/#Erdos318___P___","anchor":"Erdos318___P___","docHtml":"\n A set A : Set ℕ is said to have propery P₁ if for any nonconstant sequence\nf : A → {-1, 1}, one can always select a finite, nonempty subset S ⊆ A \\ {0} such that\n∑ n ∈ S, fₙ / n = 0. This is defined in [Sa82b].
\nℕ has property P₁. This is proved in [ErSt75].
\n Sattler proved in [Sa75] that the set of odd numbers has property P₁.
\n The set of squares does not have property P₁.
\n For any set A containing exactly one even number, A does not have property P₁. Sattler\n[Sa82] credits this observation to Erdős, who presumably found this after [ErGr80].
\n There exists a set A with positive density that does not have property P₁.\n#TODO: prove this lemma by assuming erdos_318.contain_single_even.
\n Every infinite arithmetic progression has property P₁. This is proved in [Sa82b].
\n Does the set of squares excluding 1 have property P₁?
\n Larsen [La26] proved that this set does have property P₁.
\n Defines the set of new numbers generated from a set A by the operation\n$x y - 1$ for $x \\neq y$.
"},"Erdos424.sequenceSet":{"url":"/FormalConjectures/ErdosProblems/«424»/#Erdos424___sequenceSet","anchor":"Erdos424___sequenceSet","docHtml":"\n The sequence of sets $A_n$ where $A_0 = {2, 3}$ and $A_{n+1}$ is $A_n$ union all newly\ngenerated elements.
"},"Erdos424.generatedSet":{"url":"/FormalConjectures/ErdosProblems/«424»/#Erdos424___generatedSet","anchor":"Erdos424___generatedSet","docHtml":"\n The set of integers which eventually appear in the sequence, which is the union of all $A_n$.
"},"Erdos424.erdos_424":{"url":"/FormalConjectures/ErdosProblems/«424»/#Erdos424___erdos_424","anchor":"Erdos424___erdos_424","docHtml":"\n Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \\ldots, a_n$ all possible\nvalues of $a_i a_j - 1$ with $i \\neq j$.\nIs it true that the set of integers which eventually appear has positive density?
"},"Erdos457.erdos_457":{"url":"/FormalConjectures/ErdosProblems/«457»/#Erdos457___erdos_457","anchor":"Erdos457___erdos_457","docHtml":"\n Is there some $\\epsilon > 0$ such that there are infinitely\nmany $n$ where all primes $p \\le (2 + \\epsilon) \\log n$ divide\n$$\n\\prod_{1 \\le i \\le \\log n} (n + i)?\n$$
\n\n This was formalized in Lean by Baretto and van Doorn using Aristotle.
"},"Erdos457.q":{"url":"/FormalConjectures/ErdosProblems/«457»/#Erdos457___q","anchor":"Erdos457___q","docHtml":"\n Let $q(n, k)$ denote the least prime which does not divide\n$\\prod_{1 \\le i \\le k}(n + i)$.
"},"Erdos457.erdos_457.variants.qnk":{"url":"/FormalConjectures/ErdosProblems/«457»/#Erdos457___erdos_457___variants___qnk","anchor":"Erdos457___erdos_457___variants___qnk","docHtml":"\n More generally, let $q(n, k)$ denote the least prime which\ndoes not divide $\\prod_{1 \\le i \\le k}(n + i)$. This\nproblem asks whether $q(n, \\log n) \\ge (2 + \\epsilon) \\log n$\ninfinitely often.
"},"Erdos457.erdos_457.variants.one_sub":{"url":"/FormalConjectures/ErdosProblems/«457»/#Erdos457___erdos_457___variants___one_sub","anchor":"Erdos457___erdos_457___variants___one_sub","docHtml":"\n Taking $n$ to be the product of primes\nbetween $\\log n$ and $(2 + o(1)) \\log n$ gives an example where\n$$\nq(n, \\log n) \\ge (2 + o(1)) \\log n.\n$$\nCan one prove that $q(n, \\log n) < (1 - \\epsilon) (\\log n)^2$\nfor all large $n$ and some $\\epsilon > 0$?
"},"Erdos1054.f":{"url":"/FormalConjectures/ErdosProblems/«1054»/#Erdos1054___f","anchor":"Erdos1054___f","docHtml":"\n Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest\ndivisors of $m$ for some $k\\geq 1$.
"},"Erdos1054.erdos_1054.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1054»/#Erdos1054___erdos_1054___parts___i","anchor":"Erdos1054___erdos_1054___parts___i","docHtml":"\n Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors\nof $m$ for some $k\\geq 1$. Is it true that $f(n)=o(n)$?
"},"Erdos1054.erdos_1054.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«1054»/#Erdos1054___erdos_1054___parts___ii","anchor":"Erdos1054___erdos_1054___parts___ii","docHtml":"\n Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors\nof $m$ for some $k\\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$?
"},"Erdos1054.erdos_1054.parts.iii":{"url":"/FormalConjectures/ErdosProblems/«1054»/#Erdos1054___erdos_1054___parts___iii","anchor":"Erdos1054___erdos_1054___parts___iii","docHtml":"\n Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors\nof $m$ for some $k\\geq 1$. Is it true that $\\limsup f(n)/n=\\infty$?
"},"Erdos1054.f_undefined_at_2":{"url":"/FormalConjectures/ErdosProblems/«1054»/#Erdos1054___f_undefined_at_2","anchor":"Erdos1054___f_undefined_at_2","docHtml":"\n Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors\nof $m$ for some $k\\geq 1$. Show that $f$ is undefined at $n=2$, i.e. we get the junk value $0$.
"},"Erdos1054.f_undefined_at_3":{"url":"/FormalConjectures/ErdosProblems/«1054»/#Erdos1054___f_undefined_at_3","anchor":"Erdos1054___f_undefined_at_3","docHtml":"\n Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors\nof $m$ for some $k\\geq 1$. Show that $f$ is undefined at $n=5$, i.e. we get the junk value $0$.
"},"Erdos234.erdos_234":{"url":"/FormalConjectures/ErdosProblems/«234»/#Erdos234___erdos_234","anchor":"Erdos234___erdos_234","docHtml":"\n Is it true that for all c ≥ 0, the density f c of integers for which\n(p (n + 1) - p n) / log n < c exists and is a continuous function of c?
\nBrocard's Problem\nDoes $n! + 1 = m^2$ have integer solutions other than $n = 4, 5, 7$?
"},"Erdos289.erdos_289":{"url":"/FormalConjectures/ErdosProblems/«289»/#Erdos289___erdos_289","anchor":"Erdos289___erdos_289","docHtml":"\n Is it true that, for all sufficiently large $k$, there exists finite intervals\n$I_1, \\dotsc, I_k \\subset \\mathbb{N}$ with $|I_i| \\geq 2$ for $1 \\leq i \\leq k$ such that\n$$\n1 = \\sum_{i=1}^k \\sum_{n \\in I_i} \\frac{1}{n}.\n$$
"},"Erdos357.HasDistinctSums":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___HasDistinctSums","anchor":"Erdos357___HasDistinctSums","docHtml":"\n
\n Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.
"},"Erdos357.erdos_357.parts.i":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___parts___i","anchor":"Erdos357___erdos_357___parts___i","docHtml":"\n Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. Is $f(n)=o(n)$?
"},"Erdos357.erdos_357.parts.ii.bigO_version":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___parts___ii___bigO_version","anchor":"Erdos357___erdos_357___parts___ii___bigO_version","docHtml":"\n Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nHow does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $g = O(f)$ ?
"},"Erdos357.erdos_357.parts.ii.bigO_version_symm":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___parts___ii___bigO_version_symm","anchor":"Erdos357___erdos_357___parts___ii___bigO_version_symm","docHtml":"\n Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nHow does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $f = O(g)$ ?
"},"Erdos357.erdos_357.parts.ii.bigTheta_version":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___parts___ii___bigTheta_version","anchor":"Erdos357___erdos_357___parts___ii___bigTheta_version","docHtml":"\n Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nHow does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $f = \\Theta(g)$ ?
"},"Erdos357.erdos_357.parts.ii.littleO_version":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___parts___ii___littleO_version","anchor":"Erdos357___erdos_357___parts___ii___littleO_version","docHtml":"\n Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nHow does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $g = o(f)$ ?
"},"Erdos357.erdos_357.parts.ii.littleO_version_symm":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___parts___ii___littleO_version_symm","anchor":"Erdos357___erdos_357___parts___ii___littleO_version_symm","docHtml":"\n Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nHow does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $f = o(g)$ ?
"},"Erdos357.erdos_357.variants.weisenberg":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___variants___weisenberg","anchor":"Erdos357___erdos_357___variants___weisenberg","docHtml":"\n Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nIt is known that $f(n) \\geq (2+o(1))\\sqrt{n}$.\nSource: See comment by Desmond Weisenberg here: https://www.erdosproblems.com/forum/thread/357.
"},"Erdos357.erdos_357.variants.infinite_set_lower_density":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___variants___infinite_set_lower_density","anchor":"Erdos357___erdos_357___variants___infinite_set_lower_density","docHtml":"\n Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct.\nThen $A$ has lower density 0.
"},"Erdos357.erdos_357.variants.infinite_set_density":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___variants___infinite_set_density","anchor":"Erdos357___erdos_357___variants___infinite_set_density","docHtml":"\n Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct.\nThen it is conjectured that $A$ has density 0.
"},"Erdos357.erdos_357.variants.infinite_set_sum":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___variants___infinite_set_sum","anchor":"Erdos357___erdos_357___variants___infinite_set_sum","docHtml":"\n Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct.\nThen it is conjectured that the sum $\\sum_k \\frac{1}{a_k}$ converges.
"},"Erdos357.g":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___g","anchor":"Erdos357___g","docHtml":"\n Let $g(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1, \\dotsc, a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.
"},"Erdos357.erdos_357.variants.hegyvari":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___variants___hegyvari","anchor":"Erdos357___erdos_357___variants___hegyvari","docHtml":"\n Let $g(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1, \\dotsc, a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. It is known that\n$$\\left(\\frac 1 3 + o(1) \\right)n \\leq g(n) \\leq \\left(\\frac 2 3 + o(1) \\right)n.$$
"},"Erdos357.h":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___h","anchor":"Erdos357___h","docHtml":"\n Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.
"},"Erdos357.erdos_357.variants.monotone.parts.i":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___variants___monotone___parts___i","anchor":"Erdos357___erdos_357___variants___monotone___parts___i","docHtml":"\n Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. Is $h(n)=o(n)$?
"},"Erdos357.erdos_357.variants.monotone.parts.ii.bigO_version":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___variants___monotone___parts___ii___bigO_version","anchor":"Erdos357___erdos_357___variants___monotone___parts___ii___bigO_version","docHtml":"\n Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. How does $h(n)$ grow?\nCan we find a (good) explicit function $g$ such that $g = O(h)$ ?
"},"Erdos357.erdos_357.variants.monotone.parts.ii.bigO_version_symm":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___variants___monotone___parts___ii___bigO_version_symm","anchor":"Erdos357___erdos_357___variants___monotone___parts___ii___bigO_version_symm","docHtml":"\n Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. How does $h(n)$ grow?\nCan we find a (good) explicit function $g$ such that $h = O(g)$ ?
"},"Erdos357.erdos_357.variants.monotone.parts.ii.bigTheta_version":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___variants___monotone___parts___ii___bigTheta_version","anchor":"Erdos357___erdos_357___variants___monotone___parts___ii___bigTheta_version","docHtml":"\n Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. How does $h(n)$ grow?\nCan we find a (good) explicit function $g$ such that $h = \\Theta(g)$ ?
"},"Erdos357.erdos_357.variants.monotone.parts.ii.littleO_version":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___variants___monotone___parts___ii___littleO_version","anchor":"Erdos357___erdos_357___variants___monotone___parts___ii___littleO_version","docHtml":"\n Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. How does $h(n)$ grow?\nCan we find a (good) explicit function $g$ such that $g = o(h)$ ?
"},"Erdos357.erdos_357.variants.monotone.parts.ii.littleO_version_symm":{"url":"/FormalConjectures/ErdosProblems/«357»/#Erdos357___erdos_357___variants___monotone___parts___ii___littleO_version_symm","anchor":"Erdos357___erdos_357___variants___monotone___parts___ii___littleO_version_symm","docHtml":"\n Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. How does $h(n)$ grow?\nCan we find a (good) explicit function $g$ such that $h = o(g)$ ?
"},"Erdos1072.f":{"url":"/FormalConjectures/ErdosProblems/«1072»/#Erdos1072___f","anchor":"Erdos1072___f","docHtml":"\n For any prime $p$, let $f(p)$ be the least integer such that $f(p)! + 1 \\equiv 0 \\mod p$.
"},"Erdos1072.erdos_1072.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1072»/#Erdos1072___erdos_1072___parts___i","anchor":"Erdos1072___erdos_1072___parts___i","docHtml":"\n Is it true that there are infinitely many $p$ for which $f(p) = p − 1$?
"},"Erdos1072.erdos_1072.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«1072»/#Erdos1072___erdos_1072___parts___ii","anchor":"Erdos1072___erdos_1072___parts___ii","docHtml":"\n Is it true that $f(p)/p \\to 0$ for $p \\to \\infty$ in a density 1 subset of the primes?
"},"Erdos1072.erdos_1072.variants.littleo":{"url":"/FormalConjectures/ErdosProblems/«1072»/#Erdos1072___erdos_1072___variants___littleo","anchor":"Erdos1072___erdos_1072___variants___littleo","docHtml":"\n Erdős, Hardy, and Subbarao [HaSu02], believed that the number of $p \\le x$ for which $f(p)=p−1$\nis $o(x/\\log x)$.
\n\n [HaSu02] Hardy, G. E. and Subbarao, M. V.,
\n Are there infinitely many integers $n, m$ such that $ϕ(n) = σ(m)$?
"},"Erdos845.erdos_845":{"url":"/FormalConjectures/ErdosProblems/«845»/#Erdos845___erdos_845","anchor":"Erdos845___erdos_845","docHtml":"\n Let $C > 0$. Is it true that the set of integers of the form $n = b_1 + \\cdots + b_t$,\nwith $b_1 < \\cdots < b_t$, where $b_i = 2^{k_i}3^{l_i}$ for $1 \\leq i\\leq t$ and\n$b_t \\leq Cb_1$ has density $0$?
\n\n van Doorn and Everts \\cite{vDEv25} have disproved this with $C=6$ - in fact, they prove that all\nintegers can be written as such a sum in which $b_t<6b_1$.
\n\n This was formalized in Lean by Alexeev using Aristotle.
"},"Erdos152.f":{"url":"/FormalConjectures/ErdosProblems/«152»/#Erdos152___f","anchor":"Erdos152___f","docHtml":"\n Define f n to be the minimum of |{s | s - 1 ∉ A + A, s ∈ A + A, s + 1 ∉ A + A}| as A\nranges over all Sidon sets of size n.
\n Must lim f n = ∞?
\n This was proved formally by the DeepMind prover agent [DM26a].
"},"Erdos152.erdos_152.variants.square":{"url":"/FormalConjectures/ErdosProblems/«152»/#Erdos152___erdos_152___variants___square","anchor":"Erdos152___erdos_152___variants___square","docHtml":"\n Must f n ≫ n ^ 2?
\n This stronger quadratic variant was also proved formally by the DeepMind prover agent [DM26b].
"},"Erdos43.f":{"url":"/FormalConjectures/ErdosProblems/«43»/#Erdos43___f","anchor":"Erdos43___f","docHtml":"\n Let $f(N)$ be the maximum possible size of a Sidon set in ${1,\\ldots,N}$.
"},"Erdos43.erdos_43.parts.i":{"url":"/FormalConjectures/ErdosProblems/«43»/#Erdos43___erdos_43___parts___i","anchor":"Erdos43___erdos_43___parts___i","docHtml":"\n If $A$ and $B$ are Sidon sets in ${1,\\ldots,N}$ with\n$(A-A)\\cap(B-B)={0}$, is it true that\n$$\\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq\\binom{f(N)}{2}+O(1)?$$
\n\n The answer is no; the Erdős Problems page notes that this follows from the solution to\nErdős Problem 42.
"},"Erdos43.erdos_43.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«43»/#Erdos43___erdos_43___parts___ii","anchor":"Erdos43___erdos_43___parts___ii","docHtml":"\n If $A$ and $B$ are equal-sized Sidon sets in ${1,\\ldots,N}$ with\n$(A-A)\\cap(B-B)={0}$, can the bound be improved to\n$$\\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\n\\leq (1-c+o(1))\\binom{f(N)}{2}$$\nfor some constant $c>0$?
\n\n The answer is no; the Erdős Problems page records a negative answer due to Barreto.
"},"Erdos1002.erdos_1002":{"url":"/FormalConjectures/ErdosProblems/«1002»/#Erdos1002___erdos_1002","anchor":"Erdos1002___erdos_1002","docHtml":"\n For any $0<\\alpha<1$, let $f(\\alpha,n)=\\frac{1}{\\log n}\\sum_{1\\leq k\\leq n}(\\tfrac{1}{2}-\n{ \\alpha k})$. Does $f(\\alpha,n)$ have an asymptotic distribution function?
\n\n In other words, is there a non-decreasing function $g$ such that $g(-\\infty)=0$, $g(\\infty)=1$,\nand $\\lim_{n\\to \\infty}\\lvert { \\alpha\\in (0,1): f(\\alpha,n)\\leq c}\\rvert=g(c)$?
"},"Erdos1002.erdos_1002.variants.kesten":{"url":"/FormalConjectures/ErdosProblems/«1002»/#Erdos1002___erdos_1002___variants___kesten","anchor":"Erdos1002___erdos_1002___variants___kesten","docHtml":"\n Kesten [Ke60] proved that if $f(\\alpha,\\beta,n)=\\frac{1}{\\log n}\\sum_{1\\leq k\\leq n}(\\tfrac{1}{2}-\n{\\beta+\\alpha k})$ then $f(\\alpha,\\beta,n)$ has asymptotic distribution function\n$g(c)=\\frac{1}{\\pi}\\int_{-\\infty}^{\\rho c}\\frac{1}{1+t^2}\\mathrm{d}t$, where $\\rho>0$ is an explicit\nconstant.
"},"Erdos273.erdos_273":{"url":"/FormalConjectures/ErdosProblems/«273»/#Erdos273___erdos_273","anchor":"Erdos273___erdos_273","docHtml":"\n Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \\geq 5$?
"},"Erdos273.erdos_273.variants.three":{"url":"/FormalConjectures/ErdosProblems/«273»/#Erdos273___erdos_273___variants___three","anchor":"Erdos273___erdos_273___variants___three","docHtml":"\n Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \\geq 3$?
"},"Erdos228.erdos_228":{"url":"/FormalConjectures/ErdosProblems/«228»/#Erdos228___erdos_228","anchor":"Erdos228___erdos_228","docHtml":"\n Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $\\pm1$, such\nthat $$\\sqrt n \\ll |P(z)| \\ll \\sqrt n$$ for all $|z|=1$, with the implied constants independent of\n$z$ and $n$?
\n\n The answer is yes, proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST19].
\n\n [BBMST19] Balister, P. and Bollob'{A}s, B. and Morris, R. and Sahasrabudhe, J. and Tiba, M.,
\n Is it true that there are only finitely many powers of $2$ which have only the digits $0$\nand $1$ when written in base $3$?
"},"Erdos406.erdos_406.variants.one_two":{"url":"/FormalConjectures/ErdosProblems/«406»/#Erdos406___erdos_406___variants___one_two","anchor":"Erdos406___erdos_406___variants___one_two","docHtml":"\n If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power\nof $2$.
"},"Erdos1067.InfinitelyEdgeConnected":{"url":"/FormalConjectures/ErdosProblems/«1067»/#Erdos1067___InfinitelyEdgeConnected","anchor":"Erdos1067___InfinitelyEdgeConnected","docHtml":"\n A graph is infinitely edge-connected if to disconnect the graph requires deleting\ninfinitely many edges. In other words, removing any finite set of edges leaves\nthe graph connected.
"},"Erdos1067.erdos_1067":{"url":"/FormalConjectures/ErdosProblems/«1067»/#Erdos1067___erdos_1067","anchor":"Erdos1067___erdos_1067","docHtml":"\n Does every graph with chromatic number $\\aleph_1$ contain an infinitely connected subgraph with\nchromatic number $\\aleph_1$?
\n\n Komjáth [Ko13] proved that it is consistent that the answer is no. This was improved by\nSoukup [So15], who constructed a counterexample using no extra set-theoretical assumptions. A\nsimpler elementary example was given by Bowler and Pitz [BoPi24].
\n\n This was formalized in Lean by Alexeev using Aristotle and Aleph Prover.
"},"Erdos1067.erdos_1067.variants.infinite_edge_connectivity":{"url":"/FormalConjectures/ErdosProblems/«1067»/#Erdos1067___erdos_1067___variants___infinite_edge_connectivity","anchor":"Erdos1067___erdos_1067___variants___infinite_edge_connectivity","docHtml":"\n Thomassen [Th17] constructed a counterexample to the version which asks for infinite\nedge-connectivity (that is, to disconnect the graph requires deleting infinitely many edges).
"},"Erdos741.erdos_741.parts.i":{"url":"/FormalConjectures/ErdosProblems/«741»/#Erdos741___erdos_741___parts___i","anchor":"Erdos741___erdos_741___parts___i","docHtml":"\n Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive density.\nCan one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$\nboth have positive density?
\n\n Note that this is using a literal interpretation of \"positive density\".
\n\n This was disproved by the DeepMind prover agent.
"},"Erdos741.erdos_741.variants.lower":{"url":"/FormalConjectures/ErdosProblems/«741»/#Erdos741___erdos_741___variants___lower","anchor":"Erdos741___erdos_741___variants___lower","docHtml":"\n Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive lower density.\nCan one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$\nboth have positive lower density?
"},"Erdos741.erdos_741.variants.upper":{"url":"/FormalConjectures/ErdosProblems/«741»/#Erdos741___erdos_741___variants___upper","anchor":"Erdos741___erdos_741___variants___upper","docHtml":"\n Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive upper density.\nCan one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$\nboth have positive upper density?
\n\n The DeepMind prover agent found a formal proof for this statement
"},"Erdos741.erdos_741.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«741»/#Erdos741___erdos_741___parts___ii","anchor":"Erdos741___erdos_741___parts___ii","docHtml":"\n Is there a basis $A$ of order $2$ such that if $A=A_1\\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$\ncannot both have bounded gaps?
\n\n This was proved by DeepMind prover agent.
"},"Erdos342.UniqueUlamSum":{"url":"/FormalConjectures/ErdosProblems/«342»/#Erdos342___UniqueUlamSum","anchor":"Erdos342___UniqueUlamSum","docHtml":"\nUniqueUlamSum a n m means that $m$ has a unique representation as $a(i) + a(j)$\nwith $i < j < n$.
\nIsUlamSequence a means that $a$ is the Ulam sequence (OEIS A002858):\n$a(0) = 1$, $a(1) = 2$, and for each $n \\geq 2$, $a(n)$ is the least integer\ngreater than $a(n-1)$ that has a unique representation as $a(i) + a(j)$\nwith $i < j < n$.
\n $a(0) = 1$ by definition.
"},"Erdos342.erdos_342.test.a1":{"url":"/FormalConjectures/ErdosProblems/«342»/#Erdos342___erdos_342___test___a1","anchor":"Erdos342___erdos_342___test___a1","docHtml":"\n $a(1) = 2$ by definition.
"},"Erdos342.erdos_342.test.a2":{"url":"/FormalConjectures/ErdosProblems/«342»/#Erdos342___erdos_342___test___a2","anchor":"Erdos342___erdos_342___test___a2","docHtml":"\n $a(2) = 3$: the only pair $(i,j)$ with $i < j < 2$ is $(0,1)$, giving $1 + 2 = 3$.
"},"Erdos342.erdos_342.test.a3":{"url":"/FormalConjectures/ErdosProblems/«342»/#Erdos342___erdos_342___test___a3","anchor":"Erdos342___erdos_342___test___a3","docHtml":"\n $a(3) = 4$: among sums $> 3$ with a unique representation from ${1,2,3}$,\nthe smallest is $4 = 1 + 3$. The candidate $5 = 2 + 3$ is ruled out by minimality since\n$4$ has a unique representation.
"},"Erdos342.erdos_342.parts.i":{"url":"/FormalConjectures/ErdosProblems/«342»/#Erdos342___erdos_342___parts___i","anchor":"Erdos342___erdos_342___parts___i","docHtml":"\n Do infinitely many pairs $(a, a+2)$ occur in Ulam's sequence?
"},"Erdos342.erdos_342.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«342»/#Erdos342___erdos_342___parts___ii","anchor":"Erdos342___erdos_342___parts___ii","docHtml":"\n Does Ulam's sequence eventually have periodic differences? That is, is $a(n+1) - a(n)$ eventually periodic?
"},"Erdos342.erdos_342.parts.iii":{"url":"/FormalConjectures/ErdosProblems/«342»/#Erdos342___erdos_342___parts___iii","anchor":"Erdos342___erdos_342___parts___iii","docHtml":"\n Part (iii), is the density of the sequence 0?
"},"Erdos14.nonUniqueSumCount":{"url":"/FormalConjectures/ErdosProblems/«14»/#Erdos14___nonUniqueSumCount","anchor":"Erdos14___nonUniqueSumCount","docHtml":"\n The number of integers in ${1,\\ldots,N}$ which are not representable in exactly one way\nas the sum of two elements from $A$ (either because they are not representable at all, or\nbecause they are representable in more than one way).
"},"Erdos14.almostSquareRoot":{"url":"/FormalConjectures/ErdosProblems/«14»/#Erdos14___almostSquareRoot","anchor":"Erdos14___almostSquareRoot"},"Erdos14.squareRoot":{"url":"/FormalConjectures/ErdosProblems/«14»/#Erdos14___squareRoot","anchor":"Erdos14___squareRoot"},"Erdos14.erdos_14.parts.i":{"url":"/FormalConjectures/ErdosProblems/«14»/#Erdos14___erdos_14___parts___i","anchor":"Erdos14___erdos_14___parts___i","docHtml":"\n Let $A ⊆ \\mathbb{N}$. Let $B ⊆ \\mathbb{N}$ be the set of integers which are representable\nin exactly one way as the sum of two elements from $A$. Is it true that for all\n$\\epsilon > 0$ and large $N$, $|{1,\\ldots,N} \\setminus B| \\gg_\\epsilon N^{1/2 - \\epsilon}$?
"},"Erdos14.erdos_14.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«14»/#Erdos14___erdos_14___parts___ii","anchor":"Erdos14___erdos_14___parts___ii","docHtml":"\n Is it possible that $|{1,\\ldots,N} \\setminus B| = o(N^\\frac{1}{2})$?
"},"Erdos1003.erdos_1003":{"url":"/FormalConjectures/ErdosProblems/«1003»/#Erdos1003___erdos_1003","anchor":"Erdos1003___erdos_1003","docHtml":"\n Are there infinitely many solutions to $\\phi(n) = \\phi(n+1)$, where $\\phi$ is the Euler totient\nfunction?
"},"Erdos1003.erdos_1003.variants.Icc":{"url":"/FormalConjectures/ErdosProblems/«1003»/#Erdos1003___erdos_1003___variants___Icc","anchor":"Erdos1003___erdos_1003___variants___Icc","docHtml":"\n Erdős [Er85e] says that, presumably, for every $k \\geq 1$ the equation\n$$\\phi(n) = \\phi(n+1) = \\cdots = \\phi (n+k)$$ has infinitely many solutions.
\n\n [Er85e] Erdős, P.,
\n Erdős, Pomerance, and Sárközy [EPS87] proved that for all large $x$, the number\nof $n \\leq x$ with $\\phi(n) = \\phi(n+1)$ is at most $$\\frac{x}{\\exp((\\log x)^{1/3})}$$.
\n\n [EPS87] Erd\\H os, Paul and Pomerance, Carl and S'ark\"ozy, Andr'as,
\n Are there $n$ such that there is a covering system with moduli the divisors of $n$ which is 'as\ndisjoint as possible'?
\n\n That is, for all $d\\mid n$ with $d>1$ there is an associated $a_d$ such that every integer is\ncongruent to some $a_d\\pmod{d}$, and if there is some integer $x$ with\n$$x\\equiv a_d\\pmod{d}\\textrm{ and }x\\equiv a_{d'}\\pmod{d'}$$then $(d,d')=1$.
\n\n The density of such $n$ is zero. Erdős and Graham believed that no such $n$ exist.
\n\n Adenwalla [Ad25] has proved there are no such $n$.
\n\n This was formalized by van Doorn in Lean using Aristotle.
"},"Erdos239.erdos_239":{"url":"/FormalConjectures/ErdosProblems/«239»/#Erdos239___erdos_239","anchor":"Erdos239___erdos_239","docHtml":"\n Let $f:\\mathbb{N}\\to {-1,1}$ be a multiplicative function. Is it true that\n$$ \\lim_{N\\to \\infty}\\frac{1}{N}\\sum_{n\\leq N}f(n)$$ always exists?
\n\n The answer is yes, as proved by Wirsing [Wi67], and generalised by Halász [Ha68].
"},"Erdos139.r":{"url":"/FormalConjectures/ErdosProblems/«139»/#Erdos139___r","anchor":"Erdos139___r","docHtml":"\n
\nErdős Problem 139:\nLet $r_k(N)$ be the size of the largest subset of ${1,...,N}$ which does not contain a non-trivial\n$k$-term arithmetic progression. Prove that $r_k(N) = o(N)$.
"},"Erdos1139.erdos_1139":{"url":"/FormalConjectures/ErdosProblems/«1139»/#Erdos1139___erdos_1139","anchor":"Erdos1139___erdos_1139","docHtml":"\n Let $1\\leq u_1 < u_2 < \\cdots$ be the sequence of integers with at most $2$ prime factors.\nIs it true that $$\\limsup_{k \\to \\infty} \\frac{u_{k+1}-u_k}{\\log k}=\\infty?$$
"},"Erdos1196.IsPrimitive":{"url":"/FormalConjectures/ErdosProblems/«1196»/#Erdos1196___IsPrimitive","anchor":"Erdos1196___IsPrimitive","docHtml":"\n A set is primitive if no non-associated elements of the set divide each other.
"},"Erdos1196.erdos_1196":{"url":"/FormalConjectures/ErdosProblems/«1196»/#Erdos1196___erdos_1196","anchor":"Erdos1196___erdos_1196","docHtml":"\n Let $k_1 \\geq k_2 \\geq 3$. Are there only finitely many $n_2\\geq n_1 + k_1$\nsuch that\n$$\n\\prod_{1\\leq i\\leq k_1}(n_1 + i)\\ \\text{and}\\ \\prod_{1\\leq j\\leq k_2} (n_2 + j)\n$$\nhave the same prime factors?
"},"Erdos931.erdos_931.variants.additional_condition":{"url":"/FormalConjectures/ErdosProblems/«931»/#Erdos931___erdos_931___variants___additional_condition","anchor":"Erdos931___erdos_931___variants___additional_condition","docHtml":"\n Erdős thought perhaps if the two products have the same factors then\n$n_2 > 2(n_1 + k_1)$.\nIt is an open question whether this is true when allowing a finite number of counterexamples.
"},"Erdos931.erdos_931.variants.additional_condition_nonempty":{"url":"/FormalConjectures/ErdosProblems/«931»/#Erdos931___erdos_931___variants___additional_condition_nonempty","anchor":"Erdos931___erdos_931___variants___additional_condition_nonempty","docHtml":"\n In fact there exist counterexamples, like this one found by AlphaProof.
"},"Erdos931.erdos_931.variants.exists_prime":{"url":"/FormalConjectures/ErdosProblems/«931»/#Erdos931___erdos_931___variants___exists_prime","anchor":"Erdos931___erdos_931___variants___exists_prime","docHtml":"\n Erdős was unable to prove that if the two products have the same factors\nthen there must exist a prime between $n_1$ and $n_2$.
"},"Erdos352.erdos_352":{"url":"/FormalConjectures/ErdosProblems/«352»/#Erdos352___erdos_352","anchor":"Erdos352___erdos_352","docHtml":"\n Is there some $c > 0$ such that every measurable $A \\subseteq \\mathbb{R}^2$ of measure $\\geq c$\ncontains the vertices of a triangle of area 1?
"},"Erdos522.KacCoefficients":{"url":"/FormalConjectures/ErdosProblems/«522»/#Erdos522___KacCoefficients","anchor":"Erdos522___KacCoefficients","docHtml":"\n A sequence of S of a field k is a countably infinite sequence\nof independent random variables, each uniformly distributed over S.
\n Such a sequence determines a n for each n, which is the random\npolynomial given by KacCoefficients.polynomial.
\n A sequence of S of a field k is a countably infinite sequence\nof independent random variables, each uniformly distributed over S.
\n Such a sequence determines a n for each n, which is the random\npolynomial given by KacCoefficients.polynomial.
\n A sequence of S of a field k is a countably infinite sequence\nof independent random variables, each uniformly distributed over S.
\n Such a sequence determines a n for each n, which is the random\npolynomial given by KacCoefficients.polynomial.
\n A sequence of S of a field k is a countably infinite sequence\nof independent random variables, each uniformly distributed over S.
\n Such a sequence determines a n for each n, which is the random\npolynomial given by KacCoefficients.polynomial.
\n The random polynomial associated to a sequence c : KacCoefficients S Ω μ of Kac coefficients\ngiven by ∑ i ∈ Finset.range (n + 1), c i z^i.
\n The random multiset of roots associated to a Kac polynomial
"},"Erdos522.KacCoefficients.numRootsInUnitDisk":{"url":"/FormalConjectures/ErdosProblems/«522»/#Erdos522___KacCoefficients___numRootsInUnitDisk","anchor":"Erdos522___KacCoefficients___numRootsInUnitDisk","docHtml":"\n Counts the number of roots of a Kac polynomial in the unit disk with multiplicity.
"},"Erdos522.erdos_522":{"url":"/FormalConjectures/ErdosProblems/«522»/#Erdos522___erdos_522","anchor":"Erdos522___erdos_522","docHtml":"\n Let $f(z)=\\sum_{0\\leq k\\leq n} \\epsilon_k z^k$ be a random polynomial, where\n$\\epsilon_k\\in {-1,1}$ independently uniformly at random for $0\\leq k\\leq n$.
\n\n Is it true that, if $R_n$ is the number of roots of $f(z)$ in\n${ z\\in \\mathbb{C} : \\lvert z\\rvert \\leq 1}$, then\n$$\n\\frac{R_n}{n/2}\\to 1\n$$\nalmost surely?
\n\n There is some ambiguity as to whether the intended coefficient set is ${-1, 1}$ or ${0, 1}$,\nsee erdos_522.variants.zero_one for the alternate version.
\n Let $f(z)=\\sum_{0\\leq k\\leq n} \\epsilon_k z^k$ be a random polynomial, where\n$\\epsilon_k\\in {0,1}$ independently uniformly at random for $0\\leq k\\leq n$.
\n\n Is it true that, if $R_n$ is the number of roots of $f(z)$ in\n${ z\\in \\mathbb{C} : \\lvert z\\rvert \\leq 1}$, then\n$$\n\\frac{R_n}{n/2}\\to 1\n$$\nalmost surely?
"},"Erdos522.erdos_522.variants.number_real_roots":{"url":"/FormalConjectures/ErdosProblems/«522»/#Erdos522___erdos_522___variants___number_real_roots","anchor":"Erdos522___erdos_522___variants___number_real_roots","docHtml":"\n Erdős and Offord showed that the number of real roots of a random degree n polynomial with ±1\ncoefficients is (2/π+o(1))log n.
\n Yakir proved that almost all Kac polynomials have n/2+O(n^(9/10)) many roots in {z∈C:|z|≤1}.
\n We define a Lucas sequence to be a Fibonacci sequence with arbitrary starting points\nL 0 and L 1.
\n TODO: There seems to be multiple definitions in the literature, some of which also\nallow coefficients in the reccurence relation. For now this simple definition has been\nchosen as it agrees best with the Erdős problem in this same file.\nHowever before moving this into ForMathlib one should make a concious decision about\nwhich definition to choose.
\n Is there an infinite Lucas sequence $a_0, a_1, \\ldots$ where $a_{n+2} = a_{n+1} + a_n$ for\n$n \\ge 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every\nterm of the sequence?
"},"Erdos962.Erdos962Prop":{"url":"/FormalConjectures/ErdosProblems/«962»/#Erdos962___Erdos962Prop","anchor":"Erdos962___Erdos962Prop","docHtml":"\nErdos962Prop n k : there exists $m \\le n$ such that each of\n$m+1, \\ldots, m+k$ has a prime divisor strictly larger than $k$.
\n Let $k(n)$ be the maximal $k$ such that there exists $m \\le n$ with\n$m+1, \\ldots, m+k$ each divisible by a prime $> k$.
"},"Erdos962.erdos_962":{"url":"/FormalConjectures/ErdosProblems/«962»/#Erdos962___erdos_962","anchor":"Erdos962___erdos_962","docHtml":"\n Main conjecture:
\n\n $\\log k(n) \\le (\\log n)^{(1/2 + o(1))}$
"},"Erdos962.erdos_962.variants.tang_lower_bound":{"url":"/FormalConjectures/ErdosProblems/«962»/#Erdos962___erdos_962___variants___tang_lower_bound","anchor":"Erdos962___erdos_962___variants___tang_lower_bound","docHtml":"\n Tang's lower bound [Tang]:
\n\n $\\log k(n) \\ge (1/\\sqrt{2} - o(1)) * \\sqrt{\\log n * \\log \\log n}$
"},"Erdos962.erdos_962.variants.tao_upper_bound":{"url":"/FormalConjectures/ErdosProblems/«962»/#Erdos962___erdos_962___variants___tao_upper_bound","anchor":"Erdos962___erdos_962___variants___tao_upper_bound","docHtml":"\n Tao's upper bound [Tao]:
\n\n $k(n) \\le (1 + o(1)) * n^{1/2}$
"},"Erdos109.erdos_109":{"url":"/FormalConjectures/ErdosProblems/«109»/#Erdos109___erdos_109","anchor":"Erdos109___erdos_109","docHtml":"\n Any $A\\subseteq \\mathbb{N}$ of positive upper density contains a sumset $B+C$ where both $B$ and $C$\nare infinite.
\n\n The Erdős sumset conjecture. Proved by Moreira, Richter, and Robertson [MRR19].
"},"Erdos99.HasMinDist1":{"url":"/FormalConjectures/ErdosProblems/«99»/#Erdos99___HasMinDist1","anchor":"Erdos99___HasMinDist1","docHtml":"\n A set has minimum distance $1$ if all pairwise distances are at least $1$,\nand the minimum is achieved.
"},"Erdos99.FormsEquilateralTriangle":{"url":"/FormalConjectures/ErdosProblems/«99»/#Erdos99___FormsEquilateralTriangle","anchor":"Erdos99___FormsEquilateralTriangle","docHtml":"\n Three points form an equilateral triangle of side length 1.
"},"Erdos99.erdos_99":{"url":"/FormalConjectures/ErdosProblems/«99»/#Erdos99___erdos_99","anchor":"Erdos99___erdos_99","docHtml":"\n For sufficiently large n, is it the case that any set of n points with minimum distance $1$\nthat minimizes diameter must contain an equilateral triangle of side length 1?
"},"Erdos108.erdos_108":{"url":"/FormalConjectures/ErdosProblems/«108»/#Erdos108___erdos_108","anchor":"Erdos108___erdos_108","docHtml":"\n For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r)\ncontains a subgraph of girth ≥ r and chromatic number ≥ k?
"},"Erdos501.erdos_501":{"url":"/FormalConjectures/ErdosProblems/«501»/#Erdos501___erdos_501","anchor":"Erdos501___erdos_501","docHtml":"\n For every $x \\in \\mathbb{R}$ let $A_x \\subset \\mathbb{R}$ be a bounded set with outer measure\n$< 1$. Must there exist an infinite independent set, that is, some infinite $X \\subseteq\n\\mathbb{R}$ such that $x \\notin A_y$ for all $x \\neq y \\in X$?
\n\n If the sets $A_x$ are closed and have measure $< 1$, then must there exist an independent set\nof size $3$?
\n\n Known results: Erdős–Hajnal [ErHa60] proved the existence of arbitrarily large finite\nindependent sets. Hechler [He72] showed the answer is no assuming the continuum\nhypothesis.
"},"Erdos501.erdos_501.variants.erdosHajnal_finite":{"url":"/FormalConjectures/ErdosProblems/«501»/#Erdos501___erdos_501___variants___erdosHajnal_finite","anchor":"Erdos501___erdos_501___variants___erdosHajnal_finite","docHtml":"\nErdős–Hajnal (1960): arbitrarily large finite independent sets exist.
\n\n For every n : ℕ and every family A : ℝ → Set ℝ of bounded sets with Lebesgue\nouter measure < 1, there exists a finite independent set of size at least n.
\n This was proved by Erdős and Hajnal [ErHa60].
"},"Erdos501.erdos_501.variants.hechler_CH":{"url":"/FormalConjectures/ErdosProblems/«501»/#Erdos501___erdos_501___variants___hechler_CH","anchor":"Erdos501___erdos_501___variants___hechler_CH","docHtml":"\nHechler (1972) [He72]: the answer to the main question is NO, assuming the continuum\nhypothesis.
\n\n Assuming CH (ℵ₁ = 𝔠), there exists a family A : ℝ → Set ℝ of bounded sets with\nLebesgue outer measure < 1 for which no infinite independent set exists.
\nClosed sets case: existence of an independent set of size 3.
\n\n If the sets A x are closed with Lebesgue measure < 1, must there exist an\nindependent set of size 3?
\n This is implied by the stronger theorem of Newelski–Pawlikowski–Seredyński [NPS87] below;\nGladysz [Gl62] earlier proved the existence of an independent set of size 2.
"},"Erdos501.erdos_501.variants.newelski_pawlikowski_seredynski":{"url":"/FormalConjectures/ErdosProblems/«501»/#Erdos501___erdos_501___variants___newelski_pawlikowski_seredynski","anchor":"Erdos501___erdos_501___variants___newelski_pawlikowski_seredynski","docHtml":"\nNewelski–Pawlikowski–Seredyński (1987) [NPS87]: infinite independent set in the closed case.
\n\n If all the sets A x are closed with Lebesgue measure < 1, then there is an\ninfinite independent set. This gives a strong affirmative answer to the second\nquestion of Problem 501.
\nGladysz (1962) [Gl62]: independent set of size 2 in the closed case.
\n\n If all the sets A x are closed with Lebesgue measure < 1, then there exist two\ndistinct reals x y such that x ∉ A y and y ∉ A x.
\n This is a weaker result proved by Gladysz before the full Newelski–Pawlikowski–\nSeredyński theorem [NPS87].
"},"Erdos501.erdos_501.variants.singleton_independent":{"url":"/FormalConjectures/ErdosProblems/«501»/#Erdos501___erdos_501___variants___singleton_independent","anchor":"Erdos501___erdos_501___variants___singleton_independent","docHtml":"\nTrivial lower bound: a single-element set is always independent.
\n\n For any family A, any singleton {x} is vacuously independent: there are no two\ndistinct elements.
\nTwo-element sets: independent iff mutual non-membership.
\n\n A two-element set {x, y} (with x ≠ y) is independent for A if and only if\nx ∉ A y and y ∉ A x.
\n The constant family A x = ∅ satisfies all hypotheses of the main problem:\neach A x is bounded (the empty set is bounded) and has Lebesgue outer measure 0 < 1.\nMoreover, all of ℝ is an independent set, showing the conclusion holds trivially.
\n This demonstrates that the hypotheses are non-vacuous: the family A x = ∅ is a valid\ninput to the theorem, and ℝ (which is infinite) witnesses the conclusion.
\n A singleton {0} is an independent set for any family A : ℝ → Set ℝ,\nas witnessed by erdos_501.variants.singleton_independent.
\n Two reals form an independent set for the empty family A _ = ∅:\nneither 0 nor 1 belongs to ∅, so both conditions of pair_independent_iff hold.
\n The hypothesis volume.toOuterMeasure (A x) < 1 is strictly satisfied when\nA x = {x} (a singleton), since Lebesgue measure of a singleton is 0.
\n The boundary case: the measure condition < 1 is sharp. An interval of length ≥ 1\nhas Lebesgue measure ≥ 1, so it would fail the hypothesis. Here [0, 1] has measure exactly 1.
\n Let $r\\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at\nleast one colour missing on the edges of the induced $K_{r+1}$.
\n\n In other words, there is no balanced colouring.
\n\n A conjecture of Erdős and Gyárfás [ErGy99].
"},"Erdos617.erdos_617.variants.r_eq_3":{"url":"/FormalConjectures/ErdosProblems/«617»/#Erdos617___erdos_617___variants___r_eq_3","anchor":"Erdos617___erdos_617___variants___r_eq_3","docHtml":"\n Erdős and Gyárfás [ErGy99] proved the conjecture for $r=3$.
"},"Erdos617.erdos_617.variants.r_eq_4":{"url":"/FormalConjectures/ErdosProblems/«617»/#Erdos617___erdos_617___variants___r_eq_4","anchor":"Erdos617___erdos_617___variants___r_eq_4","docHtml":"\n Erdős and Gyárfás [ErGy99] proved the conjecture for $r=4$.
"},"Erdos617.erdos_617.variants.r2":{"url":"/FormalConjectures/ErdosProblems/«617»/#Erdos617___erdos_617___variants___r2","anchor":"Erdos617___erdos_617___variants___r2","docHtml":"\n Erdős and Gyárfás [ErGy99] showed this property fails for infinitely many $r$ if we replace $r^2+1$\nby $r^2$.
"},"Erdos354.FloorMultiples":{"url":"/FormalConjectures/ErdosProblems/«354»/#Erdos354___FloorMultiples","anchor":"Erdos354___FloorMultiples","docHtml":"\n The sequence ⌊a⌋, ⌊γ * a⌋, ⌊γ ^ 2 * a⌋, ..., ⌊γ ^ i * a⌋, ....
\n The sequence ⌊a⌋, ⌊b⌋, ⌊γ * a⌋, ⌊γ * b⌋, ... ⌊γ ^ i * a⌋, ⌊γ ^ i * b⌋, ...
\n Let $\\alpha,\\beta\\in \\mathbb{R}_{>0}$ such that $\\alpha/\\beta$ is irrational. Is\n$${ \\lfloor \\alpha\\rfloor,\\lfloor \\gamma\\alpha\\rfloor,\\lfloor \\gamma^2\\alpha\\rfloor,\\ldots}\\cup\n{ \\lfloor \\beta\\rfloor,\\lfloor \\gamma\\beta\\rfloor,\\lfloor \\gamma^2\\beta\\rfloor,\\ldots}$$ complete?
"},"Erdos354.erdos_354.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«354»/#Erdos354___erdos_354___parts___ii","anchor":"Erdos354___erdos_354___parts___ii","docHtml":"\n Let $\\alpha,\\beta\\in \\mathbb{R}_{>0}$ such that $\\alpha/\\beta$ is irrational. Is\n$${ \\lfloor \\alpha\\rfloor,\\lfloor \\gamma\\alpha\\rfloor,\\lfloor \\gamma^2\\alpha\\rfloor,\\ldots}\\cup\n{ \\lfloor \\beta\\rfloor,\\lfloor \\gamma\\beta\\rfloor,\\lfloor \\gamma^2\\beta\\rfloor,\\ldots}$$ complete?
"},"Erdos125.erdos_125":{"url":"/FormalConjectures/ErdosProblems/«125»/#Erdos125___erdos_125","anchor":"Erdos125___erdos_125","docHtml":"\n Case 3:\nDoes $A + B$ have positive upper and lower density that are equal?\nThis is the literal interpretation of \"positive density\" which was falsified.
"},"Erdos125.erdos_125.variants.positive_lower_density":{"url":"/FormalConjectures/ErdosProblems/«125»/#Erdos125___erdos_125___variants___positive_lower_density","anchor":"Erdos125___erdos_125___variants___positive_lower_density","docHtml":"\n Literature question:\nDoes $A + B$ have positive lower density?
\n\n This has been falsified.
"},"Erdos125.erdos_125.variants.positive_upper_density":{"url":"/FormalConjectures/ErdosProblems/«125»/#Erdos125___erdos_125___variants___positive_upper_density","anchor":"Erdos125___erdos_125___variants___positive_upper_density","docHtml":"\n Literature question:\nDoes $A + B$ have positive upper density?
"},"Erdos125.erdos_125.variants.zero_density":{"url":"/FormalConjectures/ErdosProblems/«125»/#Erdos125___erdos_125___variants___zero_density","anchor":"Erdos125___erdos_125___variants___zero_density","docHtml":"\n Case 1:\nDoes $A + B$ have zero upper and lower density?
"},"Erdos125.erdos_125.variants.zero_lower_positive_upper_density":{"url":"/FormalConjectures/ErdosProblems/«125»/#Erdos125___erdos_125___variants___zero_lower_positive_upper_density","anchor":"Erdos125___erdos_125___variants___zero_lower_positive_upper_density","docHtml":"\n Case 2:\nDoes $A + B$ have zero lower density, but positive upper density?
"},"Erdos125.erdos_125.variants.positive_unequal_density":{"url":"/FormalConjectures/ErdosProblems/«125»/#Erdos125___erdos_125___variants___positive_unequal_density","anchor":"Erdos125___erdos_125___variants___positive_unequal_density","docHtml":"\n Case 4:\nDoes $A + B$ have positive upper and lower density that are unequal?
\n\n This follows from the disproof erdos_125.variants.positive_lower_density above.
\n Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a for some a in A\nand n ≥ 0.
\n Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a\nfor some a in A and n ≥ 0. What is the smallest possible value of\nlim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2)?
\n Erdos observed that this value is finite and > 1.
"},"Erdos33.erdos_33.variants.vanDoorn":{"url":"/FormalConjectures/ErdosProblems/«33»/#Erdos33___erdos_33___variants___vanDoorn","anchor":"Erdos33___erdos_33___variants___vanDoorn","docHtml":"\n The smallest possible value of lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2)\nis at most 2φ^(5/2) ≈ 6.66, with φ equal to the golden ratio. Proven by\nWouter van Doorn.
\n The length of a subset $s$ of $\\mathbb{C}$ is defined to be its 1-dimensional\nHausdorff measure $\\mathcal{H}^1(s)$.
"},"Erdos1041.exists_connected_component_contains_two_roots":{"url":"/FormalConjectures/ErdosProblems/«1041»/#Erdos1041___exists_connected_component_contains_two_roots","anchor":"Erdos1041___exists_connected_component_contains_two_roots","docHtml":"\nErdős–Herzog–Piranian Component Lemma (Metric Properties of Polynomials, 1958):\nIf $f$ is a monic degree $n$ polynomial with all roots in the unit disk,\nthen some connected component\nof ${z \\mid |f(z)| < 1}$ contains at least two roots with multiplicity.
\n\n See p. 139, above Problem 5:\n[EHP58] Erdős, P. and Herzog, F. and Piranian, G.,
\n Let\n$$ f(z) = \\prod_{i=1}^{n} (z - z_i) \\in \\mathbb{C}[x] $$\nwith $|z_i| < 1$ for all $i$.
\n\n Conjecture: Must there always exist a path of length less than 2 in\n$$ { z \\in \\mathbb{C} \\mid |f(z)| < 1 } $$\nwhich connects two of the roots of $f$?
"},"Erdos1077.erdos_1077":{"url":"/FormalConjectures/ErdosProblems/«1077»/#Erdos1077___erdos_1077","anchor":"Erdos1077___erdos_1077","docHtml":"\n We call a graph $D$-balanced (or $D$-almost-regular) if the maximum degree is at most $D$ times the\nminimum degree.
\n\n Let $ε, α > 0$ and $D$ and $n$ be sufficiently large. If $G$ is a graph on $n$ vertices with at\nleast $n^{1+α}$ edges, then must $G$ contain a $D$-balanced subgraph on $m > n^{1-α}$ vertices with\nat least $εm^{1+α}$ edges?
"},"Erdos124.sumsOfDistinctPowers":{"url":"/FormalConjectures/ErdosProblems/«124»/#Erdos124___sumsOfDistinctPowers","anchor":"Erdos124___sumsOfDistinctPowers","docHtml":"\n The set of integers which are the sum of distinct powers d ^ i with i ≥ k.
\n Let $3 \\le d_1 < d_2 < \\dots < d_r$ be integers such that\n$$\\sum_{1 \\le i \\le r}\\frac 1{d_i - 1} \\ge 1.$$\nCan all sufficiently large integers be written as a sum of the shape $\\sum_i c_ia_i$\nwhere $c_i \\in {0, 1}$ and $a_i$ has only the digits $0, 1$ when written in base $d_i$?
\n\n Conjectured by Erdős [Er97], solved by Boris Alexeev using Aristotle.
"},"Erdos124.erdos124.ne_zero":{"url":"/FormalConjectures/ErdosProblems/«124»/#Erdos124___erdos124___ne_zero","anchor":"Erdos124___erdos124___ne_zero","docHtml":"\n Let $k \\ne 0$ and $3\\leq d_1 < d_2 < \\cdots < d_r$ be integers of gcd equal to $1$ such that\n$$\\sum_{1 \\le i \\le r}\\frac 1{d_i - 1} \\ge 1.$$\nCan all sufficiently large integers be written as a sum of the shape $\\sum_i c_ia_i$\nwhere $c_i \\in {0, 1}$ and $a_i$ is divisible by $d_i ^ k$ and has only the digits $0, 1$ when\nwritten in base $d_i$?
\n\n Conjectured by Burr, Erdős, Graham, and Li [BEGL96]
"},"Erdos124.erdos124.ne_zero_three_four_seven":{"url":"/FormalConjectures/ErdosProblems/«124»/#Erdos124___erdos124___ne_zero_three_four_seven","anchor":"Erdos124___erdos124___ne_zero_three_four_seven","docHtml":"\n All sufficiently large integers can be written as $a + b + c$ where $a$ has only the digits $0, 1$\nin base $3$, $b$ only the digits $0, 1$ in base $4$, $c$ only the digits $0, 1$ in base $7$.
\n\n Provee by Burr, Erdős, Graham, and Li [BEGL96]
"},"Erdos124.erdos124.converse":{"url":"/FormalConjectures/ErdosProblems/«124»/#Erdos124___erdos124___converse","anchor":"Erdos124___erdos124___converse","docHtml":"\n Let $3\\leq d_1 < d_2 < \\cdots < d_r$ be integers such that all sufficiently large integers can be\nwritten as a sum of the shape $\\sum_i c_ia_i$ where $c_i \\in {0, 1}$ and $a_i$ has only the digits\n$0, 1$ when written in base $d_i$. Then\n$$\\sum_{1 \\le i \\le r}\\frac 1{d_i - 1} \\ge 1.$$
\n\n Reported by Burr, Erdős, Graham, and Li [BEGL96] as an observation of Pomerance
"},"Erdos124.erdos124.melfi_construction":{"url":"/FormalConjectures/ErdosProblems/«124»/#Erdos124___erdos124___melfi_construction","anchor":"Erdos124___erdos124___melfi_construction","docHtml":"\n For any $\\varepsilon > 0$, there exists an infinite sequence $2 \\le d_0 < d_1 < \\dots$ such\nthat all sufficiently large integer can be written as $\\sum_{i \\in I} a_i$ where $a_i$ has only\nthe digits $0, 1$ when written in base $d_i$,\nbut $\\sum_{i \\in I} \\frac 1{d_i - 1} \\le \\varepsilon$.
\n\n Proved by Melfi [Me04]
"},"Erdos477.erdos_477":{"url":"/FormalConjectures/ErdosProblems/«477»/#Erdos477___erdos_477","anchor":"Erdos477___erdos_477","docHtml":"\n Is there a polynomial $f:\\mathbb{Z}\\to \\mathbb{Z}$ of degree at least $2$ and a set\n$A\\subset \\mathbb{Z}$ such that for any $z\\in \\mathbb{Z}$ there is exactly one $a\\in A$ and\n$b\\in { f(n) : n\\in\\mathbb{Z}}$ such that $z=a+b$?
"},"Erdos477.erdos_477.variants.S_sq":{"url":"/FormalConjectures/ErdosProblems/«477»/#Erdos477___erdos_477___variants___S_sq","anchor":"Erdos477___erdos_477___variants___S_sq","docHtml":"\n There is no such $A$ for the polynomial $f(x) = X^2$.
\n\n This is shown in [Sek59].
"},"Erdos477.erdos_477.variants.degree_two_dvd_condition_b_ne_zero":{"url":"/FormalConjectures/ErdosProblems/«477»/#Erdos477___erdos_477___variants___degree_two_dvd_condition_b_ne_zero","anchor":"Erdos477___erdos_477___variants___degree_two_dvd_condition_b_ne_zero","docHtml":"\n There is no such $A$ for any polynomial $f(x) = aX^2 + bX + c$, if $a | b$\nwith $a \\ne 0$ and $b \\ne 0.\nThis was found be AlphaProof for the specific instance $X^2 - X + 1$ and then generalised.
"},"Erdos477.erdos_477.variants.X_pow_three":{"url":"/FormalConjectures/ErdosProblems/«477»/#Erdos477___erdos_477___variants___X_pow_three","anchor":"Erdos477___erdos_477___variants___X_pow_three","docHtml":"\n Probably there is no such $A$ for the polynomial $X^3$.
"},"Erdos477.erdos_477.variants.monomial":{"url":"/FormalConjectures/ErdosProblems/«477»/#Erdos477___erdos_477___variants___monomial","anchor":"Erdos477___erdos_477___variants___monomial","docHtml":"\n Probably there is no such $A$ for the polynomial $X^k$ for any $k \\ge 2$. This is asked in [Sek59].
"},"Erdos890.omegaGt":{"url":"/FormalConjectures/ErdosProblems/«890»/#Erdos890___omegaGt","anchor":"Erdos890___omegaGt","docHtml":"\nomegaGt k n counts the number of distinct prime factors of n that are strictly\ngreater than k.
\n If $\\omega_k(n)$ counts the number of distinct prime factors of $n$ which are $>k$, then is it true\nthat, for every $k\\geq 1$,\n$$\\liminf_{n\\to \\infty}\\sum_{0\\leq i < k}\\omega_k(n+i)\\leq k?$$
"},"Erdos890.erdos_890.parts.b":{"url":"/FormalConjectures/ErdosProblems/«890»/#Erdos890___erdos_890___parts___b","anchor":"Erdos890___erdos_890___parts___b","docHtml":"\n Is it true that\n$$\\limsup_{n\\to \\infty}\\left(\\sum_{0\\leq i < k}\\omega(n+i)\\right) \\frac{\\log\\log n}{\\log n}=1,$$\nwhere $\\omega$ counts the number of distinct prime factors without restriction?
"},"Erdos890.erdos_890.variants.liminf_lower_bound":{"url":"/FormalConjectures/ErdosProblems/«890»/#Erdos890___erdos_890___variants___liminf_lower_bound","anchor":"Erdos890___erdos_890___variants___liminf_lower_bound","docHtml":"\n A question of Erdős and Selfridge [ErSe67], who observe that\n$\\liminf_{n\\to \\infty}\\sum_{0\\leq i < k}\\omega(n+i)\\geq k+\\pi(k)-1$ for every $k$. This follows from\nPólya's theorem that the set of $k$-smooth integers has unbounded gaps - indeed,\n$n(n+1)\\cdots (n+k-1)$ is divisible by all primes $\\leq k$ and, provided $n$ is large, all but at\nmost one of $n,n+1,\\ldots,n+k-1$ has a prime factor $>k$ by Pólya's theorem.
"},"Erdos890.erdos_890.variants.omega_limsup":{"url":"/FormalConjectures/ErdosProblems/«890»/#Erdos890___erdos_890___variants___omega_limsup","anchor":"Erdos890___erdos_890___variants___omega_limsup","docHtml":"\n It is a classical fact that $\\limsup_{n\\to \\infty}\\omega(n)\\frac{\\log\\log n}{\\log n}=1.$
"},"Erdos1092.f":{"url":"/FormalConjectures/ErdosProblems/«1092»/#Erdos1092___f","anchor":"Erdos1092___f","docHtml":"\n $f_r(n)$ is maximal such that, if a graph $G$ on $n$ vertices has the property that every\nsubgraph $H$ on $m$ vertices has chromatic number $\\leq r$ once we remove $f_r(m)$ edges\nfrom it, then $G$ has chromatic number $\\leq r+1$.
"},"Erdos1092.f_asymptotic_2":{"url":"/FormalConjectures/ErdosProblems/«1092»/#Erdos1092___f_asymptotic_2","anchor":"Erdos1092___f_asymptotic_2","docHtml":"\n Is $f_2(n) \\gg n$? Disproved by Rödl, who showed $f_r(n) = o(n)$ for all fixed $r \\geq 2$.
"},"Erdos1092.f_asymptotic_general":{"url":"/FormalConjectures/ErdosProblems/«1092»/#Erdos1092___f_asymptotic_general","anchor":"Erdos1092___f_asymptotic_general","docHtml":"\n Is $f_r(n) \\gg_r n$ for all $r$? Disproved by Rödl, who showed $f_r(n) = o(n)$ for all fixed\n$r \\geq 2$.
"},"Erdos248.erdos_248":{"url":"/FormalConjectures/ErdosProblems/«248»/#Erdos248___erdos_248","anchor":"Erdos248___erdos_248","docHtml":"\n Are there infinitely many $n$ such that $\\omega(n + k) \\ll k$ for all $k \\geq 1$?\nHere $\\omega(n)$ is the number of distinct prime divisors of $n$.
"},"Erdos680.erdos_680.parts.i":{"url":"/FormalConjectures/ErdosProblems/«680»/#Erdos680___erdos_680___parts___i","anchor":"Erdos680___erdos_680___parts___i","docHtml":"\n Is it true that, for all sufficiently large $n$, there exists some $k$ such that\n$$\np(n+k)>k^2+1,\n$$\nwhere $p(m)$ denotes the least prime factor of $m$?
"},"Erdos680.erdos_680.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«680»/#Erdos680___erdos_680___parts___ii","anchor":"Erdos680___erdos_680___parts___ii","docHtml":"\n Can one prove this is false if we replace $k^2+1$ by $e^{(1+\\epsilon)\\sqrt{k}}+C_\\epsilon$, for all\n$\\epsilon>0$, where $C_\\epsilon>0$ is some constant?
"},"Erdos678.lcmInterval_lt_example1":{"url":"/FormalConjectures/ErdosProblems/«678»/#Erdos678___lcmInterval_lt_example1","anchor":"Erdos678___lcmInterval_lt_example1","docHtml":"\n The referee of [Er79] found the example $M(96, 7) > M(104, 8)$, showing that there are cases where\n$M(n, k) > M(m, k + 1)$ with $m \\geq n + k$.\n[Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
"},"Erdos678.lcmInterval_lt_example2":{"url":"/FormalConjectures/ErdosProblems/«678»/#Erdos678___lcmInterval_lt_example2","anchor":"Erdos678___lcmInterval_lt_example2","docHtml":"\n The referee of [Er79] found the example $M(132, 7) > M(139, 8)$, showing that there are cases where\n$M(n, k) > M(m, k + 1)$ with $m \\geq n + k$.\n[Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
"},"Erdos678.lcmInterval_lt_example3":{"url":"/FormalConjectures/ErdosProblems/«678»/#Erdos678___lcmInterval_lt_example3","anchor":"Erdos678___lcmInterval_lt_example3","docHtml":"\n Cambie [Ca24] found the example $M(52, 7) > M(62, 8)$.\n[Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).
"},"Erdos678.lcmInterval_lt_example4":{"url":"/FormalConjectures/ErdosProblems/«678»/#Erdos678___lcmInterval_lt_example4","anchor":"Erdos678___lcmInterval_lt_example4","docHtml":"\n Cambie [Ca24] found the example $M(36, 8) > M(48, 9)$.\n[Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).
"},"Erdos678.erdos_678":{"url":"/FormalConjectures/ErdosProblems/«678»/#Erdos678___erdos_678","anchor":"Erdos678___erdos_678","docHtml":"\n Write $M(n, k)$ be the least common multiple of ${n+1, \\dotsc, n+k}$.\nLet $k$ be sufficiently large. Are there infinitely many $m, n$ with $m \\geq n + k$ such that\n$$\nM(n, k) > M(m, k + 1)\n$$?\nThe answer is yes, as proved in a strong form by Cambie [Ca24].\n[Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).
\n\n This was formalized in Lean by Alexeev using Aristotle, conditional on asymptotic estimates for the\nprime counting function (specifically pi_alt from the PNT+ project).\nSee the formal proof.
\n The set of k for which a / b can be expressed as a sum of k distinct unit fractions.
\n Let $$N(a, b)$$, denoted here by smallestCollection a b be the minimal k such that there\nexist integers $1 < n_1 < n_2 < \\dots < n_k$ with\n$$\\frac{a}{b} = \\sum_{i=1}^k \\frac{1}{n_i}$$
\n Write $$N(b) = max_{1 \\leq a < b} N(a, b)$$.
"},"Erdos304.erdos_304.variants.upper_1950":{"url":"/FormalConjectures/ErdosProblems/«304»/#Erdos304___erdos_304___variants___upper_1950","anchor":"Erdos304___erdos_304___variants___upper_1950","docHtml":"\n In 1950, Erdős [Er50c] proved the upper bound $$N(b) \\ll \\log b / \\log \\log b$$.\n[Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \\ldots + {1}/{x_n} =A/B$ egyenlet eg'{E}sz sz'{A}m'{u} megold'{A}sairól. Mat. Lapok (1950), 192-210.
"},"Erdos304.erdos_304.variants.lower_1950":{"url":"/FormalConjectures/ErdosProblems/«304»/#Erdos304___erdos_304___variants___lower_1950","anchor":"Erdos304___erdos_304___variants___lower_1950","docHtml":"\n In 1950, Erdős [Er50c] proved the lower bound $$\\log \\log b \\ll N(b)$$.\n[Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \\ldots + {1}/{x_n} =A/B$ egyenlet eg'{E}sz sz'{A}m'{u} megold'{A}sairól. Mat. Lapok (1950), 192-210.
"},"Erdos304.erdos_304.variants.upper_1985":{"url":"/FormalConjectures/ErdosProblems/«304»/#Erdos304___erdos_304___variants___upper_1985","anchor":"Erdos304___erdos_304___variants___upper_1985","docHtml":"\n In 1985 Vose [Vo85] proved the upper bound $$N(b) \\ll \\sqrt{\\log b}$$.\n[Vo85] Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24.
"},"Erdos304.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«304»/#Erdos304___upper_bound","anchor":"Erdos304___upper_bound","docHtml":"\n Is it true that $$N(b) \\ll \\log \\log b$$?
"},"Erdos245.erdos_245":{"url":"/FormalConjectures/ErdosProblems/«245»/#Erdos245___erdos_245","anchor":"Erdos245___erdos_245","docHtml":"\n Let $A\\subseteq\\mathbb{N}$ be an infinite set such that $|A\\cap {1, ..., N}| = o(N)$.\nIs it true that\n$$\n\\limsup_{N\\to\\infty}\\frac{|(A + A)\\cap {1, ..., N}|}{|A \\cap {1, ..., N}|} \\geq 3?\n$$
\n\n The answer is yes, proved by Freiman [Fr73].
\n\n [Fr73] Fre\\u{\\i}man, G. A.,
\n Let $A\\subseteq\\mathbb{N}$ be an infinite set such that $|A\\cap {1, ..., N}| = o(N)$.\nThen\n$$\n\\limsup_{N\\to\\infty}\\frac{|(A + A)\\cap {1, ..., N}|}{|A \\cap {1, ..., N}|} \\geq 2.\n$$
"},"Erdos888.RequiredCondition":{"url":"/FormalConjectures/ErdosProblems/«888»/#Erdos888___RequiredCondition","anchor":"Erdos888___RequiredCondition","docHtml":"\n Condition on the sets A appearing in Erdős 888. Namely, let A be a subset\nof {1,...,n} such that if a ≤ b ≤ c ≤ d ∈ A and abcd square then ad=bc.
\n Proposition that for a specific n an A with the above defined condition\nand cardinality k exists.
\n What is the size of the largest subset A of {1,...,n} such that if\na ≤ b ≤ c ≤ d ∈ A and abcd square then ad=bc
\n|A|=o(n).
\n The primes show that |A| ≫ n/log n is possible.
\n Let q : ℕ → ℕ be a strictly increasing sequence of primes such that\nq (n + 2) - q (n + 1) ≥ q (n + 1) - q n. Must lim q n / (n ^ 2) = ∞?
\n Let q : ℕ → ℕ be a strictly increasing sequence of primes such that\nq (n + 2) - q (n + 1) ≥ q (n + 1) - q n. Then liminf q n / (n ^ 2) > 0.352, and this is proved in\n[Ri76].
\n Let $F(n) := \\max{m + p(m) \\mid \\textrm{$m < n$ composite}}}$ where $p(m)$ is the least\nprime divisor of $m$.
"},"Erdos385.trivial_ub":{"url":"/FormalConjectures/ErdosProblems/«385»/#Erdos385___trivial_ub","anchor":"Erdos385___trivial_ub","docHtml":"\n Note that trivially $F(n) \\leq n + \\sqrt{n}$.
"},"Erdos385.erdos_385.parts.i":{"url":"/FormalConjectures/ErdosProblems/«385»/#Erdos385___erdos_385___parts___i","anchor":"Erdos385___erdos_385___parts___i","docHtml":"\n Let $F(n) := \\max{m + p(m) \\mid \\textrm{$m < n$ composite}}}$ where $p(m)$ is the least\nprime divisor of $m$. Is it true that $F(n)>n$ for all sufficiently large $n$?
"},"Erdos385.erdos_385.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«385»/#Erdos385___erdos_385___parts___ii","anchor":"Erdos385___erdos_385___parts___ii","docHtml":"\n Let $F(n) := \\max{m + p(m) \\mid \\textrm{$m < n$ composite}}}$ where $p(m)$ is the least\nprime divisor of $m$. Does $F(n) - n \\to \\infty$ as $n\\to\\infty$?
"},"Erdos385.erdos_385.variants.lb":{"url":"/FormalConjectures/ErdosProblems/«385»/#Erdos385___erdos_385___variants___lb","anchor":"Erdos385___erdos_385___variants___lb","docHtml":"\n A question of Erdős, Eggleton, and Selfridge, who write that in fact it is possible that\nthis quantity is always at least $n+(1-o(1))\\sqrt{n}$
"},"Erdos410.erdos_410":{"url":"/FormalConjectures/ErdosProblems/«410»/#Erdos410___erdos_410","anchor":"Erdos410___erdos_410","docHtml":"\n Let $σ_1(n) = σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$.
\n\n Is it true that $\\lim_{k → ∞} σ_k(n)^{\\frac 1 k} = ∞$?
\n\n This is problem (iii) from\nErdos, Granville, Pomerance, Spiro\n\"On the normal behavior of the iterates of some arithmetical functions\"\n(page 169 of the book \"Analytic Number Theory\", 1990).
"},"Erdos11.erdos_11":{"url":"/FormalConjectures/ErdosProblems/«11»/#Erdos11___erdos_11","anchor":"Erdos11___erdos_11","docHtml":"\n Is every odd $n > 1$ the sum of a squarefree number and a power of 2?
"},"Erdos11.erdos_11.variants.not_four_dvd":{"url":"/FormalConjectures/ErdosProblems/«11»/#Erdos11___erdos_11___variants___not_four_dvd","anchor":"Erdos11___erdos_11___variants___not_four_dvd","docHtml":"\n Erdős often asked this under the weaker assumption that $n > 1$\nis not divisible by 4.
"},"Erdos11.erdos_11.variants.two_pow_two":{"url":"/FormalConjectures/ErdosProblems/«11»/#Erdos11___erdos_11___variants___two_pow_two","anchor":"Erdos11___erdos_11___variants___two_pow_two","docHtml":"\n Is every odd $n > 1$ the sum of a squarefree number and two powers of 2?
"},"Erdos11.erdos_11.variants.finite_bound1":{"url":"/FormalConjectures/ErdosProblems/«11»/#Erdos11___erdos_11___variants___finite_bound1","anchor":"Erdos11___erdos_11___variants___finite_bound1","docHtml":"\n Every odd $1 < n < 10^7$ is the sum of a squarefree number and a power of 2.
"},"Erdos11.erdos_11.variants.finite_bound2":{"url":"/FormalConjectures/ErdosProblems/«11»/#Erdos11___erdos_11___variants___finite_bound2","anchor":"Erdos11___erdos_11___variants___finite_bound2","docHtml":"\n Every odd $1 < n < 2^50$ is the sum of a squarefree number and a power of 2.
"},"Erdos11.erdos_11.variants.granville_soundararajan":{"url":"/FormalConjectures/ErdosProblems/«11»/#Erdos11___erdos_11___variants___granville_soundararajan","anchor":"Erdos11___erdos_11___variants___granville_soundararajan","docHtml":"\n Suppose that every odd $n$ is the sum of a squarefree number and a power of 2. Then the set of primes\n$p$ such that $2 ^ p ≡ 2 \\mod p ^ 2$ is infinite. This is Theorem 1 in [GrSo98].\n[GrSo98] Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$. The Ramanujan Journal (1998), 283-298.
"},"Erdos250.erdos_250":{"url":"/FormalConjectures/ErdosProblems/«250»/#Erdos250___erdos_250","anchor":"Erdos250___erdos_250","docHtml":"\n Is\n$$\n\\sum_{n=1}^\\infty \\frac{\\sigma(n)}{2^n}\n$$\nirrational? Here $\\sigma(n)$ is the sum of divisors function.
\n\n The answer is yes, as shown by Nesterenko [Ne96].
\n\n [Ne96] Nesterenko, Yu V.,
\n Let $A\\subseteq\\mathbb{N}$ be an infinite set such that $|A\\cap {1, ..., N}| = o(N)$.\nIs it true that\n$$\n\\limsup_{N\\to\\infty}\\frac{|(A - A)\\cap {1, ..., N}|}{|A \\cap {1, ..., N}|} = \\infty?\n$$
\n\n The answer is yes, proved by Ruzsa [Ru78].
\n\n [Ru78] Ruzsa, I. Z.,
\n $f(n)$ counts the number of solutions to $n=p+2^k$ for prime $p$ and $k\\geq 0$.
"},"Erdos236.erdos_236":{"url":"/FormalConjectures/ErdosProblems/«236»/#Erdos236___erdos_236","anchor":"Erdos236___erdos_236","docHtml":"\n Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\\geq 0$. Show that $f(n)=o(\\log n)$.
"},"Erdos1071.SegmentsDisjoint":{"url":"/FormalConjectures/ErdosProblems/«1071»/#Erdos1071___SegmentsDisjoint","anchor":"Erdos1071___SegmentsDisjoint","docHtml":"\n Two segments are disjoint if they only intersect at their endpoints (if at all).
"},"Erdos1071.erdos_1071.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1071»/#Erdos1071___erdos_1071___parts___i","anchor":"Erdos1071___erdos_1071___parts___i","docHtml":"\n Can a finite set of disjoint unit segments in a unit square be maximal?\nSolved affirmatively by [Da85], who gave an explicit construction.
\n\n This was formalized in Lean by Alexeev using Aristotle and ChatGPT.
"},"Erdos1071.erdos_1071.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«1071»/#Erdos1071___erdos_1071___parts___ii","anchor":"Erdos1071___erdos_1071___parts___ii","docHtml":"\n Is there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite?\nSolved affirmatively by [Fo99], who gave an explicit construction.
\n\n This was formalized in Lean by Alexeev using Aristotle and ChatGPT.
"},"Erdos208.erdos208.s":{"url":"/FormalConjectures/ErdosProblems/«208»/#Erdos208___erdos208___s","anchor":"Erdos208___erdos208___s","docHtml":"\n The sequence of squarefree numbers, denoted by s as in Erdős problem 208.
\n Let $s_1 < s_2 < \\dots$ be the sequence of squarefree numbers. Is it true that\nfor any $\\epsilon > 0$ and large $n$, $s_{n+1} - s_n \\ll_\\epsilon s_n^\\epsilon$?
"},"Erdos208.erdos_208.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«208»/#Erdos208___erdos_208___parts___ii","anchor":"Erdos208___erdos_208___parts___ii","docHtml":"\n Let $s_1 < s_2 < \\dots$ be the sequence of squarefree numbers. Is it true that\n$s_{n + 1} - s_n \\le (1 + o(1)) \\cdot (\\pi^2 / 6) \\cdot \\log (s_n) / \\log (\\log (s_n))$?
"},"Erdos208.erdos_208.variants.log_bound":{"url":"/FormalConjectures/ErdosProblems/«208»/#Erdos208___erdos_208___variants___log_bound","anchor":"Erdos208___erdos_208___variants___log_bound","docHtml":"\n In [Er79] Erdős says perhaps $s_{n+1} - s_n \\ll \\log s_n$, but he is 'very doubtful'.
\n\n [Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
"},"Erdos1214.erdos_1214":{"url":"/FormalConjectures/ErdosProblems/«1214»/#Erdos1214___erdos_1214","anchor":"Erdos1214___erdos_1214","docHtml":"\n Let $x,y\\geq 1$ be integers such that, for all $n\\geq 1$, the set of primes dividing $x^{n}-1$ is\nequal to the set of primes dividing $y^n-1$. Must $x=y$?
\n\n Erdős asked this at a 1988 number theory conference in Banff.
\n\n A positive answer was given by Corrales-Rodrigáñez and Schoof [CoSc97].
"},"Erdos397.erdos_397":{"url":"/FormalConjectures/ErdosProblems/«397»/#Erdos397___erdos_397","anchor":"Erdos397___erdos_397","docHtml":"\n Are there only finitely many solutions to\n$$\n\\prod_i \\binom{2m_i}{m_i}=\\prod_j \\binom{2n_j}{n_j}\n$$\nwith the $m_i,n_j$ distinct?
\n\n Somani, using ChatGPT, has given a negative answer. In fact, for any $a\\geq 2$, if $c=8a^2+8a+1$,\n$\\binom{2a}{a}\\binom{4a+4}{2a+2}\\binom{2c}{c}= \\binom{2a+2}{a+1}\\binom{4a}{2a}\\binom{2c+2}{c+1}.$\nFurther families of solutions are given in the comments by SharkyKesa.
\n\n This was earlier asked about in a [MathOverflow] question, in response to which Elkies also gave an\nalternative construction which produces solutions - at the moment it is not clear whether Elkies'\nargument gives infinitely many solutions (although Bloom believes that it can).
\n\n This was formalized in Lean by Wu using Aristotle.
"},"Erdos39.erdos_39":{"url":"/FormalConjectures/ErdosProblems/«39»/#Erdos39___erdos_39","anchor":"Erdos39___erdos_39","docHtml":"\n Is there an infinite Sidon set $A\\subset \\mathbb{N}$ such that\n$\\lvert A\\cap {1\\ldots,N}\\rvert \\gg_\\epsilon N^{1/2-\\epsilon}$\nfor all $\\varepsilon > 0$?
"},"Erdos997.IsWellDistributed":{"url":"/FormalConjectures/ErdosProblems/«997»/#Erdos997___IsWellDistributed","anchor":"Erdos997___IsWellDistributed","docHtml":"\n Call $x_1,x_2,\\ldots \\in (0,1)$ well-distributed if, for every $\\epsilon>0$, if $k$ is\nsufficiently large then, for all $n>0$ and intervals $I\\subseteq [0,1]$,\n$\\lvert # { n < m\\leq n+k : x_m\\in I} - \\lvert I\\rvert k\\rvert < \\epsilon k.$
\n\n The notion of a well-distributed sequence was introduced by Hlawka and Petersen [Hl55].
"},"Erdos997.erdos_997":{"url":"/FormalConjectures/ErdosProblems/«997»/#Erdos997___erdos_997","anchor":"Erdos997___erdos_997","docHtml":"\n Is it true that, for every $\\alpha$, the sequence ${ \\alpha p_n}$ is not well-distributed,\nif $p_n$ is the sequence of primes?
\n\n The answer is yes, by [APSSV26, Section 4]; a Lean formalisation is available in [Mo26].
"},"Erdos997.erdos_997.variants.lacunary":{"url":"/FormalConjectures/ErdosProblems/«997»/#Erdos997___erdos_997___variants___lacunary","anchor":"Erdos997___erdos_997___variants___lacunary","docHtml":"\n Erdős proved that, if $n_k$ is a lacunary sequence, then the sequence ${ \\alpha n_k}$ is not\nwell-distributed for almost all $\\alpha$.
"},"Erdos997.erdos_997.variants.irrational":{"url":"/FormalConjectures/ErdosProblems/«997»/#Erdos997___erdos_997___variants___irrational","anchor":"Erdos997___erdos_997___variants___irrational","docHtml":"\n He also claimed in [Er64b] to have proved that there exists an irrational $\\alpha$ for which\n${\\alpha p_n}$ is not well-distributed. He later retracted this claim in [Er85e], saying \"The\ntheorem is no doubt correct and perhaps will not be difficult to prove but I never was able to\nreconstruct my 'proof' which perhaps never existed.\"
\n\n The existence of such an $\\alpha$ was established by Champagne, Le, Liu, and Wooley [CLLW24].
"},"Erdos486.erdos_486":{"url":"/FormalConjectures/ErdosProblems/«486»/#Erdos486___erdos_486","anchor":"Erdos486___erdos_486","docHtml":"\n For each $n \\in \\mathbb{N}$ choose some $X_n \\subseteq \\mathbb{Z}/n\\mathbb{Z}$.\nLet $B = {m \\in \\mathbb{N} : \\forall n, m \\not\\equiv x \\pmod{n} \\text{ for all } x \\in X_n}$.\nMust $B$ have a logarithmic density?
"},"Erdos645.erdos_645":{"url":"/FormalConjectures/ErdosProblems/«645»/#Erdos645___erdos_645","anchor":"Erdos645___erdos_645","docHtml":"\n If ℕ is $2$-coloured then there must exist a monochromatic three-term arithmetic progression\n$x,x+d,x+2d$ such that $d>x$.
\n\n This was first proved by Brown and Landman [BrLa99], who in fact show that this is always possible\nwith $d>f(x)$ for any increasing function $f$.
\n\n This was formalized in Lean by Alexeev using Aristotle and ChatGPT.
"},"Erdos1210.erdos_1210":{"url":"/FormalConjectures/ErdosProblems/«1210»/#Erdos1210___erdos_1210","anchor":"Erdos1210___erdos_1210","docHtml":"\n Let $A\\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\\in A$.\nIs it true that $\\sum_{a\\in A}\\frac{1}{n-a}\\leq \\sum_{p < n}\\frac{1}{p}+O(1)$?
"},"Erdos1210.erdos_1210.variants.er80_correction":{"url":"/FormalConjectures/ErdosProblems/«1210»/#Erdos1210___erdos_1210___variants___er80_correction","anchor":"Erdos1210___erdos_1210___variants___er80_correction","docHtml":"\n In [Er80] he claims he \"did not state this quite correctly\" in [Er77c]. The problem in [Er77c] which\nErdős is presumably referring to states that if $n < q_1 < \\cdots < q_k\\leq m$ is the set of primes\nin $(n,m]$ then $\\sum \\frac{1}{q_i-n} < \\sum_{p < m-n}\\frac{1}{p}+O(1)$.
"},"Erdos509.BoundedDiscCover":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___BoundedDiscCover","anchor":"Erdos509___BoundedDiscCover","docHtml":"\n An $r$-bounded disc cover of a subset of a metric space $M$\nis an indexed family of closed discs whose radii sum to at most $r$.
"},"Erdos509.BoundedDiscCover.C":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___BoundedDiscCover___C","anchor":"Erdos509___BoundedDiscCover___C","docHtml":"\n An $r$-bounded disc cover of a subset of a metric space $M$\nis an indexed family of closed discs whose radii sum to at most $r$.
"},"Erdos509.BoundedDiscCover.R":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___BoundedDiscCover___R","anchor":"Erdos509___BoundedDiscCover___R","docHtml":"\n An $r$-bounded disc cover of a subset of a metric space $M$\nis an indexed family of closed discs whose radii sum to at most $r$.
"},"Erdos509.BoundedDiscCover.h_cover":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___BoundedDiscCover___h_cover","anchor":"Erdos509___BoundedDiscCover___h_cover","docHtml":"\n An $r$-bounded disc cover of a subset of a metric space $M$\nis an indexed family of closed discs whose radii sum to at most $r$.
"},"Erdos509.BoundedDiscCover.h_summable":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___BoundedDiscCover___h_summable","anchor":"Erdos509___BoundedDiscCover___h_summable","docHtml":"\n An $r$-bounded disc cover of a subset of a metric space $M$\nis an indexed family of closed discs whose radii sum to at most $r$.
"},"Erdos509.BoundedDiscCover.h_bdd":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___BoundedDiscCover___h_bdd","anchor":"Erdos509___BoundedDiscCover___h_bdd","docHtml":"\n An $r$-bounded disc cover of a subset of a metric space $M$\nis an indexed family of closed discs whose radii sum to at most $r$.
"},"Erdos509.BoundedDiscCover.h_pos":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___BoundedDiscCover___h_pos","anchor":"Erdos509___BoundedDiscCover___h_pos","docHtml":"\n An $r$-bounded disc cover of a subset of a metric space $M$\nis an indexed family of closed discs whose radii sum to at most $r$.
"},"Erdos509.boundedDiscCover_empty":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___boundedDiscCover_empty","anchor":"Erdos509___boundedDiscCover_empty"},"Erdos509.BoundedDiscCover.bound_nonneg_of_nonempty":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___BoundedDiscCover___bound_nonneg_of_nonempty","anchor":"Erdos509___BoundedDiscCover___bound_nonneg_of_nonempty"},"Erdos509.erdos_509":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___erdos_509","anchor":"Erdos509___erdos_509","docHtml":"\n Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. Can the set\n${z ∈ ℂ : |f(z)| ≤ 1}$\nbe covered by a set of closed discs the sum of whose radii is $≤ 2$?
"},"Erdos509.erdos_509.variants.Cartan_bound":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___erdos_509___variants___Cartan_bound","anchor":"Erdos509___erdos_509___variants___Cartan_bound","docHtml":"\n Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. Can the set\n${z ∈ ℂ : |f(z)| ≤ 1}$\nbe covered by a set of closed discs the sum of whose radii is $≤ 2e$?\nSolution: True. This is due to Cartan.\nSee
\n Let $f(z) ∈ $ℂ[z]$ be a monic non-constant polynomial. Can the set\n${z ∈ ℂ : |f(z)| ≤ 1}$\nbe covered by a set of closed discs the sum of whose radii is $≤ 2.59$?\nSolution: True. This is due to Pommerenke.
"},"Erdos509.erdos_509.variants.Pommerenke_connected":{"url":"/FormalConjectures/ErdosProblems/«509»/#Erdos509___erdos_509___variants___Pommerenke_connected","anchor":"Erdos509___erdos_509___variants___Pommerenke_connected","docHtml":"\n Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial.\nIf it is connected, can the set ${z ∈ ℂ : |f(z)| ≤ 1}$\nbe covered by a set of circles the sum of whose radii is $≤ 2$?\nSolution: True. This is due to Pommerenke.
"},"Erdos881.IsMinimalAsymptoticAddBasisOfOrder":{"url":"/FormalConjectures/ErdosProblems/«881»/#Erdos881___IsMinimalAsymptoticAddBasisOfOrder","anchor":"Erdos881___IsMinimalAsymptoticAddBasisOfOrder","docHtml":"\n We interpret \"additive basis of order k\" as an asymptotic additive basis of order k,\nusing the predicate Set.IsAsymptoticAddBasisOfOrder from additive combinatorics.
\n A k is a set A such that
\nA is an asymptotic additive basis of order k, and
\n for every infinite subset B ⊆ A, the complement A \\ B is k.
\n Let A ⊂ ℕ be an additive basis of order k which is minimal in the sense that\nif B ⊂ A is any infinite set, then A \\ B is not a basis of order k.
\n Must there exist an infinite B ⊂ A such that A \\ B\nis an additive basis of order k + 1?
\n Are there infinitely many $n$ such that, for all $k\\geq 1$\n$$\n\\tau(n + k) \\ll k?\n$$
"},"Erdos17.IsClusterPrime":{"url":"/FormalConjectures/ErdosProblems/«17»/#Erdos17___IsClusterPrime","anchor":"Erdos17___IsClusterPrime","docHtml":"\n A prime $p$ is a cluster prime if every even natural number\n$n \\le p - 3$ can be written as a difference of two primes\n$q_1 - q_2$ with $q_1, q_2 \\le p$.
"},"Erdos17.erdos_17":{"url":"/FormalConjectures/ErdosProblems/«17»/#Erdos17___erdos_17","anchor":"Erdos17___erdos_17","docHtml":"\nErdős Problem 17. Are there infinitely many cluster primes?
"},"Erdos17.clusterPrimeCount":{"url":"/FormalConjectures/ErdosProblems/«17»/#Erdos17___clusterPrimeCount","anchor":"Erdos17___clusterPrimeCount","docHtml":"\n The counting function of cluster primes $\\le n$.
"},"Erdos17.erdos_17.variants.upper_BES":{"url":"/FormalConjectures/ErdosProblems/«17»/#Erdos17___erdos_17___variants___upper_BES","anchor":"Erdos17___erdos_17___variants___upper_BES","docHtml":"\n In 1999 Blecksmith, Erdős, and Selfridge [BES99] proved the upper bound\n$$\\pi^{\\mathcal{C}}(x) \\ll_A x(\\log x)^{-A}$$ for every real $A > 0$.
\n\n [BES99] Blecksmith, Richard and Erd\\H os, Paul and Selfridge, J. L., Cluster primes. Amer. Math. Monthly (1999), 43--48.
"},"Erdos17.erdos_17.variants.upper_Elsholtz":{"url":"/FormalConjectures/ErdosProblems/«17»/#Erdos17___erdos_17___variants___upper_Elsholtz","anchor":"Erdos17___erdos_17___variants___upper_Elsholtz","docHtml":"\n In 2003, Elsholtz [El03] refined the upper bound to\n$$\\pi^{\\mathcal{C}}(x) \\ll x,\\exp!\\bigl(-c(\\log\\log x)^2\\bigr)$$\nfor every real $0 < c < 1/8$.
\n\n [El03] Elsholtz, Christian, On cluster primes. Acta Arith. (2003), 281--284.
"},"Erdos17.isClusterPrime_97_isLeast_non_cluster":{"url":"/FormalConjectures/ErdosProblems/«17»/#Erdos17___isClusterPrime_97_isLeast_non_cluster","anchor":"Erdos17___isClusterPrime_97_isLeast_non_cluster","docHtml":"\n $97$ is the smallest prime that is not a cluster prime.
"},"Erdos428.primeDensityRatio":{"url":"/FormalConjectures/ErdosProblems/«428»/#Erdos428___primeDensityRatio","anchor":"Erdos428___primeDensityRatio","docHtml":"\n The density ratio of set $A$ up to $n$ relative to the prime counting function $\\pi(n)$.
"},"Erdos428.erdos_428":{"url":"/FormalConjectures/ErdosProblems/«428»/#Erdos428___erdos_428","anchor":"Erdos428___erdos_428","docHtml":"\n Is there a set $A\\subseteq \\mathbb{N}$ such that, for infinitely many $n$, all of $n-a$\nare prime for all $a\\in A$ with $0 < a < n$ and $$\\liminf\\frac{\\lvert A\\cap [1,x]\\rvert}{\\pi(x)}>0?$$
"},"Erdos1051.GrowthCondition":{"url":"/FormalConjectures/ErdosProblems/«1051»/#Erdos1051___GrowthCondition","anchor":"Erdos1051___GrowthCondition","docHtml":"\n A sequence of integers a satisfies the growth condition if\n$\\liminf a_n^{\\frac{1}{2^n}} > 1$.
\n The series $\\sum_{n=0}^\\infty \\frac{1}{a_n \\cdot a_{n+1}}$.
"},"Erdos1051.erdos_1051":{"url":"/FormalConjectures/ErdosProblems/«1051»/#Erdos1051___erdos_1051","anchor":"Erdos1051___erdos_1051","docHtml":"\n Is it true that if $a_0 < a_1 < a_2 < \\cdots$ is a strictly increasing sequence\nof integers with $\\liminf a_n^{1/2^n} > 1$, then the series\n$\\sum_{n=0}^\\infty \\frac{1}{a_n \\cdot a_{n+1}}$ is irrational?
\n\n This was solved in the affirmative by Aletheia [Fe26]. This was extended by Barreto, Kang, Kim,\nKovač, and Zhang [BKKKZ26], who essentially give a complete answer: if $\\phi=\\frac{1+\\sqrt{5}}{2}$\nis the golden ratio and $1\\leq a_1 < a_2 < \\cdots$ is a monotonically increasing sequence of\nintegers such that $\\limsup a_n^{1/\\phi^{n}}=\\infty$ then $\\sum_{n=1}^\\infty \\frac{1}{a_na_{n+1}}$\nis irrational. Conversely, for any $1 < C < \\infty$ there exists a sequence of integers\n$1\\leq a_1<\\cdots$ such that $\\lim a_n^{1/\\phi^{n}}=C$ where this infinite sum is a rational number.
\n\n (Further, more general, results are available in [BKKKZ26].)
\n\n This was formalized in Lean by Baretto.
"},"Erdos1051.erdos_1051.variants.rapid_growth":{"url":"/FormalConjectures/ErdosProblems/«1051»/#Erdos1051___erdos_1051___variants___rapid_growth","anchor":"Erdos1051___erdos_1051___variants___rapid_growth","docHtml":"\n Erdős [Er88c] notes that if the sequence grows rapidly to infinity (specifically, if\n$a_{n+1} \\geq C \\cdot a_n^2$ for some constant $C > 0$), then the series is irrational.
"},"Erdos979.solutionSet":{"url":"/FormalConjectures/ErdosProblems/«979»/#Erdos979___solutionSet","anchor":"Erdos979___solutionSet","docHtml":"\n
\n Let $k ≥ 2$, and let $f_k(n)$ count the number of solutions to $n = p_1^k + \\dots + p_k^k$,\nwhere the $p_i$ are prime numbers. Is it true that $\\limsup f_k(n) = \\infty$?
"},"Erdos979.erdos_979.variants.k2":{"url":"/FormalConjectures/ErdosProblems/«979»/#Erdos979___erdos_979___variants___k2","anchor":"Erdos979___erdos_979___variants___k2","docHtml":"\n Erdős [Er37b] proved that if $f_2(n)$ counts the number of solutions to $n = p_1^2 + p_2^2$, where $p_1$ and $p_2$ are prime numbers, then $\\limsup f_2(n) = \\infty$.
\n\n [Er37b] Erdős, Paul, On the Sum and Difference of Squares of Primes. J. London Math. Soc. (1937), 133--136.
"},"Erdos979.erdos_979.variants.k3":{"url":"/FormalConjectures/ErdosProblems/«979»/#Erdos979___erdos_979___variants___k3","anchor":"Erdos979___erdos_979___variants___k3","docHtml":"\n Erdős (unpublished)
"},"Erdos1073.A":{"url":"/FormalConjectures/ErdosProblems/«1073»/#Erdos1073___A","anchor":"Erdos1073___A","docHtml":"\n Let $A(x)$ count the number of composite $u < x$ such that $n!+1 \\equiv 0 (\\mod u)$ for some $n$.
"},"Erdos1073.erdos_1073":{"url":"/FormalConjectures/ErdosProblems/«1073»/#Erdos1073___erdos_1073","anchor":"Erdos1073___erdos_1073","docHtml":"\n Is it true that $A(x) \\le x^{o(1)}$?
"},"Erdos92.maxEquidistantPointsAt":{"url":"/FormalConjectures/ErdosProblems/«92»/#Erdos92___maxEquidistantPointsAt","anchor":"Erdos92___maxEquidistantPointsAt","docHtml":"\n For a given point x and a set of other points, this function finds the maximum number of points\nthat lie on a single circle centered at x. It does this by grouping the other points by their\ndistance to x and finding the size of the largest group.
\n This property holds for a set of points A if every point x in A has at least k other\npoints from A that are equidistant from x.
\n The set of all possible values k for which there exists a set of n points\nsatisfying the hasMinEquidistantProperty k. The function f(n) will be the supremum of this set.
\n A sanity check to ensure the set of possible f(n) values is bounded above. A trivial bound is\nn-1, since any point can have at most n-1 other points equidistant from it.\nThis ensures sSup is well-defined.
\n Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\\mathbb^2$\nin which every $x \\in A$ has at least $f(n)$ points in $A$ equidistant from $x$.
"},"Erdos92.erdos_92.variants.weak":{"url":"/FormalConjectures/ErdosProblems/«92»/#Erdos92___erdos_92___variants___weak","anchor":"Erdos92___erdos_92___variants___weak","docHtml":"\n Is it true that $f(n)\\leq n^{o(1)}$?
"},"Erdos92.erdos_92.variants.strong":{"url":"/FormalConjectures/ErdosProblems/«92»/#Erdos92___erdos_92___variants___strong","anchor":"Erdos92___erdos_92___variants___strong","docHtml":"\n Or even $f(n) < n^{c/\\log\\log n}$ for some constant $c > 0$?
"},"Erdos269.HasPrimeFactorsIn":{"url":"/FormalConjectures/ErdosProblems/«269»/#Erdos269___HasPrimeFactorsIn","anchor":"Erdos269___HasPrimeFactorsIn","docHtml":"\n A positive integer $n$ has all its prime factors in the set $P$.\nBy convention, $1$ satisfies this for any $P$ as it has no prime divisors.
"},"Erdos269.a":{"url":"/FormalConjectures/ErdosProblems/«269»/#Erdos269___a","anchor":"Erdos269___a","docHtml":"\n The infinite, strictly increasing sequence ${a_0, a_1, \\dots}$ of integers\nwhose prime factors all belong to $P$.
"},"Erdos269.partialLcm":{"url":"/FormalConjectures/ErdosProblems/«269»/#Erdos269___partialLcm","anchor":"Erdos269___partialLcm","docHtml":"\n The $n$-th partial least common multiple, $[a_0, \\dots, a_{n-1}]$, which is\nthe LCM of the first $n$ integers in the sequence.
"},"Erdos269.series":{"url":"/FormalConjectures/ErdosProblems/«269»/#Erdos269___series","anchor":"Erdos269___series","docHtml":"\n The sum $\\sum_{n=1}^\\infty \\frac{1}{[a_0,\\ldots,a_{n - 1}]}$.
"},"Erdos269.erdos_269.variants.rational":{"url":"/FormalConjectures/ErdosProblems/«269»/#Erdos269___erdos_269___variants___rational","anchor":"Erdos269___erdos_269___variants___rational","docHtml":"\n Let $P$ be a finite set of primes with $|P| \\ge 2$ and let\n${a_1 < a_2 < \\dots}$ be the set of positive integers whose prime factors\nare all in $P$. Is the sum\n$$ \\sum_{n=1}^\\infty \\frac{1}{[a_1,\\ldots,a_n]} $$\nrational?
"},"Erdos269.erdos_269.variants.irrational":{"url":"/FormalConjectures/ErdosProblems/«269»/#Erdos269___erdos_269___variants___irrational","anchor":"Erdos269___erdos_269___variants___irrational","docHtml":"\n Let $P$ be a finite set of primes with $|P| \\ge 2$ and let\n${a_1 < a_2 < \\dots}$ be the set of positive integers whose prime factors\nare all in $P$. Is the sum\n$$ \\sum_{n=1}^\\infty \\frac{1}{[a_1,\\ldots,a_n]} $$\nirrational?
"},"Erdos269.erdos_269.variants.infinite":{"url":"/FormalConjectures/ErdosProblems/«269»/#Erdos269___erdos_269___variants___infinite","anchor":"Erdos269___erdos_269___variants___infinite","docHtml":"\n This theorem addresses the case where the set of primes $P$ is infinite. In this case the sum is\nirrational.
"},"Erdos347.erdos_347":{"url":"/FormalConjectures/ErdosProblems/«347»/#Erdos347___erdos_347","anchor":"Erdos347___erdos_347","docHtml":"\n Is there a sequence $A={a_1\\leq a_2\\leq \\cdots}$ of integers with\n$$\\lim \\frac{a_{n+1}}{a_n}=2$$\nsuch that\n$$P(A')= \\left{\\sum_{n\\in B}n : B\\subseteq A'\\textrm{ finite }\\right}$$\nhas density $1$ for every cofinite subsequence $A'$ of $A$?
\n\n This has been solved in the affirmative by ebarschkis in the comments (based on idea of Tao and\nvan Doorn, also in the comments).
\n\n Thos was formalized in Lean by Barschkis using Aristotle.
"},"Erdos1093.deficiency":{"url":"/FormalConjectures/ErdosProblems/«1093»/#Erdos1093___deficiency","anchor":"Erdos1093___deficiency","docHtml":"\n If defined, the deficiency is the count of $0 \\le i < k$ such that $n - i$ is $k$-smooth.
"},"Erdos1093.erdos_1093.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1093»/#Erdos1093___erdos_1093___parts___i","anchor":"Erdos1093___erdos_1093___parts___i","docHtml":"\n Are there infinitely many binomial coefficients with deficiency 1?
"},"Erdos1093.erdos_1093.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«1093»/#Erdos1093___erdos_1093___parts___ii","anchor":"Erdos1093___erdos_1093___parts___ii","docHtml":"\n Are there only finitely many binomial coefficients with deficiency > 1?
"},"Erdos1080.IsBipartition":{"url":"/FormalConjectures/ErdosProblems/«1080»/#Erdos1080___IsBipartition","anchor":"Erdos1080___IsBipartition","docHtml":"\nIsBipartition G X Y means that X and Y form a bipartition of the vertices of G.
\n Let $G$ be a bipartite graph on $n$ vertices such that one part has $\\lfloor n^{2/3}\\rfloor$\nvertices. Is there a constant $c>0$ such that if $G$ has at least $cn$ edges then $G$ must\ncontain a $C_6$?
\n\n The answer is no, as shown by De Caen and Székely [DeSz92], who in fact show a stronger result.\nLet $f(n,m)$ be the maximum number of edges of a bipartite graph between $n$ and $m$ vertices which\ndoes not contain either a $C_4$ or $C_6$. A positive answer to this question would then imply\n$f(n,\\lfloor n^{2/3}\\rfloor)\\ll n$. De Caen and Székely prove\n$n^{10/9}\\gg f(n,\\lfloor n^{2/3}\\rfloor) \\gg n^{58/57+o(1)}$ for $m\\sim n^{2/3}$. They also prove\nmore generally that, for $n^{1/2}\\leq m\\leq n$, $f(n,m) \\ll (nm)^{2/3},$ which was also proved by\nFaudree and Simonovits.
\n\n This was formalized in Lean by Alexeev using Aristotle.
"},"Erdos51.erdos_51":{"url":"/FormalConjectures/ErdosProblems/«51»/#Erdos51___erdos_51","anchor":"Erdos51___erdos_51","docHtml":"\n Is there an infinite set $A \\subset \\mathbb{N}$ such that for every $a \\in A$,\nthere is an integer n such that $\\phi(n)=a$, and\nyet if $n_a$ is the smallest such integer, then $\\frac{n_a}{a} → \\infty$ as $a → ∞$?
"},"Erdos920.f":{"url":"/FormalConjectures/ErdosProblems/«920»/#Erdos920___f","anchor":"Erdos920___f","docHtml":"\n $f_k(n)$ is the maximum possible chromatic number of a graph with $n$ vertices\nwhich contains no $K_k$.
"},"Erdos920.erdos_920":{"url":"/FormalConjectures/ErdosProblems/«920»/#Erdos920___erdos_920","anchor":"Erdos920___erdos_920","docHtml":"\n Is it true that, for $k\\geq 4$, $f_k(n) \\gg \\frac{n^{1-\\frac{1}{k-1}}}{(\\log n)^{c_k}}$ for some\nconstant $c_k>0$?
"},"Erdos920.erdos_920.variants.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«920»/#Erdos920___erdos_920___variants___upper_bound","anchor":"Erdos920___erdos_920___variants___upper_bound","docHtml":"\n Graver and Yackel [GrYa68] proved that\n$f_k(n) \\ll \\left(n\\frac{\\log\\log n}{\\log n}\\right)^{1-\\frac{1}{k-1}}.$
"},"Erdos920.erdos_920.variants.k_eq_3":{"url":"/FormalConjectures/ErdosProblems/«920»/#Erdos920___erdos_920___variants___k_eq_3","anchor":"Erdos920___erdos_920___variants___k_eq_3","docHtml":"\n It is known that $f_3(n)\\asymp (n/\\log n)^{1/2}$ (see [erdosproblems.com/1104]).
"},"Erdos920.erdos_920.variants.lower_bound_f4":{"url":"/FormalConjectures/ErdosProblems/«920»/#Erdos920___erdos_920___variants___lower_bound_f4","anchor":"Erdos920___erdos_920___variants___lower_bound_f4","docHtml":"\n The lower bound $R(4,m) \\gg m^3/(\\log m)^4$ of Mattheus and Verstraete [MaVe23]\n(see [erdosproblems.com/166]) implies $f_4(n) \\gg \\frac{n^{2/3}}{(\\log n)^{4/3}}$.
"},"Erdos920.erdos_920.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«920»/#Erdos920___erdos_920___variants___lower_bound","anchor":"Erdos920___erdos_920___variants___lower_bound","docHtml":"\n A positive answer to this question would follow from [erdosproblems.com/986]. The known bounds for\nthat problem imply $f_k(n) \\gg \\frac{n^{1-\\frac{2}{k+1}}}{(\\log n)^{c_k}}.$
"},"Erdos945.τ":{"url":"/FormalConjectures/ErdosProblems/«945»/#Erdos945______","anchor":"Erdos945______","docHtml":"\n
\n [ErMi52] Erdős, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}. Proc. London Math. Soc. (3) (1952), 257--271.
\n\n Let $F(x)$ be the maximal $k$ such that there exist $n+1, \\dots, n+k \\le x$\nwith $τ(n+1), \\dots, τ(n+k)$ all distinct, where $τ(m)$ counts the divisors of $m$.
"},"Erdos945.Erdos945Prop":{"url":"/FormalConjectures/ErdosProblems/«945»/#Erdos945___Erdos945Prop","anchor":"Erdos945___Erdos945Prop"},"Erdos945.erdos_945":{"url":"/FormalConjectures/ErdosProblems/«945»/#Erdos945___erdos_945","anchor":"Erdos945___erdos_945","docHtml":"\n Is it true that $F(x) \\leq (\\log x)^{O(1)}$?
"},"Erdos945.Erdos945Constant":{"url":"/FormalConjectures/ErdosProblems/«945»/#Erdos945___Erdos945Constant","anchor":"Erdos945___Erdos945Constant"},"Erdos945.erdos_945.variants.constant":{"url":"/FormalConjectures/ErdosProblems/«945»/#Erdos945___erdos_945___variants___constant","anchor":"Erdos945___erdos_945___variants___constant","docHtml":"\n Is there a constant $C > 0$ such that, for all large $x$, every interval $[x, x+(\\log x)C]$\ncontains two integers with the same number of divisors?
"},"Erdos945.erdos_945.variants.equivalence":{"url":"/FormalConjectures/ErdosProblems/«945»/#Erdos945___erdos_945___variants___equivalence","anchor":"Erdos945___erdos_945___variants___equivalence","docHtml":"\n The two ways of phrasing the conjecture are equivalent.
"},"Erdos945.erdos_945.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«945»/#Erdos945___erdos_945___variants___lower_bound","anchor":"Erdos945___erdos_945___variants___lower_bound","docHtml":"\n Erdős and Mirsky [ErMi52] proved that $\\frac{(\\log x)^{1/2}}{\\log\\log x}\\ll F(x)$.
"},"Erdos945.erdos_945.variants.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«945»/#Erdos945___erdos_945___variants___upper_bound","anchor":"Erdos945___erdos_945___variants___upper_bound","docHtml":"\n Erdős and Mirsky [ErMi52] proved that $\\log F(x) \\ll \\frac{(\\log x)^{1/2}}$.
"},"Erdos258.erdos_258":{"url":"/FormalConjectures/ErdosProblems/«258»/#Erdos258___erdos_258","anchor":"Erdos258___erdos_258","docHtml":"\n Let $a_n \\to \\infty$ be a sequence of non-zero natural numbers. Is\n$\\sum_n \\frac{d(n)}{(a_1 ... a_n)}$ irrational, where $d(n)$ is the number of divisors of $n$?
\n\n This was proved affirmatively by Chojecki and GPT-5.4 Pro [Ch26], and formalised in Lean\nby ster-oc [St26].
"},"Erdos258.erdos_258.variants.monotone":{"url":"/FormalConjectures/ErdosProblems/«258»/#Erdos258___erdos_258___variants___monotone","anchor":"Erdos258___erdos_258___variants___monotone","docHtml":"\n Let $2 \\leq a_1 \\leq a_2 \\leq \\cdots$ be a monotone sequence with $a_n \\to \\infty$.\nIs $\\sum_n \\frac{d(n)}{a_1 \\cdots a_n}$ irrational, where $d(n)$ is the number of divisors of $n$?
\n\n Solution: True (proved by Erdős and Straus [ErSt71], Lemma 2.2 and Theorem 2.13).
"},"Erdos258.erdos_258.variants.constant":{"url":"/FormalConjectures/ErdosProblems/«258»/#Erdos258___erdos_258___variants___constant","anchor":"Erdos258___erdos_258___variants___constant","docHtml":"\n Is $\\sum_n \\frac{d(n)}{t^n}$ irrational, where $t ≥ 2$ is an integer.
\n\n Solution: True (proved by Erdős, see Erdős Problems website)
"},"Erdos891.erdos_891":{"url":"/FormalConjectures/ErdosProblems/«891»/#Erdos891___erdos_891","anchor":"Erdos891___erdos_891","docHtml":"\n Let $2=p_1 < p_2 < \\cdots$ be the primes and $k\\geq 2$. Is it true that, for all sufficiently large\n$n$, there must exist an integer in $[n,n+p_1\\cdots p_k)$ with $>k$ many prime factors?
"},"Erdos891.erdos_891.variants.schinzel":{"url":"/FormalConjectures/ErdosProblems/«891»/#Erdos891___erdos_891___variants___schinzel","anchor":"Erdos891___erdos_891___variants___schinzel","docHtml":"\n Schinzel deduced from Pólya's theorem [Po18] (that the sequence of $k$-smooth integers has unbounded\ngaps) that this is true with $p_1\\cdots p_k$ replaced by $p_1\\cdots p_{k-1}p_{k+1}$.
"},"Erdos891.erdos_891.variants.case_k_2":{"url":"/FormalConjectures/ErdosProblems/«891»/#Erdos891___erdos_891___variants___case_k_2","anchor":"Erdos891___erdos_891___variants___case_k_2","docHtml":"\n This is unknown even for $k=2$ - that is, is it true that in every interval of $6$\n(sufficiently large) consecutive integers there must exist one with at least $3$ prime factors?
"},"Erdos891.erdos_891.variants.weisenberg":{"url":"/FormalConjectures/ErdosProblems/«891»/#Erdos891___erdos_891___variants___weisenberg","anchor":"Erdos891___erdos_891___variants___weisenberg","docHtml":"\n Weisenberg has observed that Dickson's conjecture implies the answer is no if we replace\n$p_1\\cdots p_k$ with $p_1\\cdots p_k-1$. Indeed, let $L_k$ be the lowest common multiple of all\nintegers at most $p_1\\cdots p_k$. By Dickson's conjecture [Wikipedia], there are infinitely many\n$n'$ such that $\\frac{L_k}{m}n'+1$ is prime for all $1\\leq m < p_1\\cdots p_k$. It follows that,\nif $n=L_kn'+1$, then all integers in $[n,n+p_1\\cdots p_k-1)$ have at most $k$ prime factors.
"},"Erdos938.erdos_938":{"url":"/FormalConjectures/ErdosProblems/«938»/#Erdos938___erdos_938","anchor":"Erdos938___erdos_938","docHtml":"\n Let $A={n_1 < n_2 < \\cdots}$ be the sequence of powerful numbers (if $p\\mid n$ then $p^2\\mid n$).\nAre there only finitely many three-term progressions of consecutive terms $n_k,n_{k+1},n_{k+2}$?
"},"Erdos1063.n":{"url":"/FormalConjectures/ErdosProblems/«1063»/#Erdos1063___n","anchor":"Erdos1063___n","docHtml":"\n Let $n_k$ be the least $n \\ge 2k$ such that all but one of the integers $n - i$ with\n$0 \\le i < k$ divide $\\binom{n}{k}$.
"},"Erdos1063.erdos_1063.better_upper":{"url":"/FormalConjectures/ErdosProblems/«1063»/#Erdos1063___erdos_1063___better_upper","anchor":"Erdos1063___erdos_1063___better_upper","docHtml":"\n Estimate $n_k$ by finding a better upper bound.
"},"Erdos1063.erdos_1063.variants.exists_exception":{"url":"/FormalConjectures/ErdosProblems/«1063»/#Erdos1063___erdos_1063___variants___exists_exception","anchor":"Erdos1063___erdos_1063___variants___exists_exception","docHtml":"\n Erdős and Selfridge noted that, for $n \\ge 2k$ with $k \\ge 2$, at least one of the numbers\n$n - i$ for $0 \\le i < k$ fails to divide $\\binom{n}{k}$ ([ErSe83]).
"},"Erdos1063.erdos_1063.variants.small_values":{"url":"/FormalConjectures/ErdosProblems/«1063»/#Erdos1063___erdos_1063___variants___small_values","anchor":"Erdos1063___erdos_1063___variants___small_values","docHtml":"\n The initial values satisfy $n_2 = 4$, $n_3 = 6$, $n_4 = 9$, and $n_5 = 12$ ([Gu04], Problem B31).
"},"Erdos1063.erdos_1063.variants.monier_upper_bound":{"url":"/FormalConjectures/ErdosProblems/«1063»/#Erdos1063___erdos_1063___variants___monier_upper_bound","anchor":"Erdos1063___erdos_1063___variants___monier_upper_bound","docHtml":"\n Monier observed that $n_k \\le k!$ for $k \\ge 3$ ([Mo85]).\nTODO: Find reference
"},"Erdos1063.erdos_1063.variants.cambie_upper_bound":{"url":"/FormalConjectures/ErdosProblems/«1063»/#Erdos1063___erdos_1063___variants___cambie_upper_bound","anchor":"Erdos1063___erdos_1063___variants___cambie_upper_bound","docHtml":"\nCambie observed the improved bound\n$n_k \\le k \\cdot \\operatorname{lcm}(1, \\dotsc, k - 1)$.
"},"Erdos1063.erdos_1063.variants.exp_upper_bound":{"url":"/FormalConjectures/ErdosProblems/«1063»/#Erdos1063___erdos_1063___variants___exp_upper_bound","anchor":"Erdos1063___erdos_1063___variants___exp_upper_bound","docHtml":"\n The least common multiple bound implies $n_k \\le \\exp((1 + o(1))k)$.
"},"Erdos402.erdos_402":{"url":"/FormalConjectures/ErdosProblems/«402»/#Erdos402___erdos_402","anchor":"Erdos402___erdos_402","docHtml":"\n Prove that, for any finite set $A\\subset\\mathbb{N}$, there exist $a, b\\in A$ such\nthat\n$$\n\\gcd(a, b)\\leq a/|A|.\n$$
"},"Erdos402.erdos_402.variants.equality":{"url":"/FormalConjectures/ErdosProblems/«402»/#Erdos402___erdos_402___variants___equality","anchor":"Erdos402___erdos_402___variants___equality","docHtml":"\n A conjecture of Graham [Gr70], who also conjectured that (assuming $A$ itself\nhas no common divisor) the only cases where equality is achieved are when\n$A = {1, \\dots, n}$ or $A = {L/1, \\dots, L/n}$ (where $L = \\operatorname{lcm}(1, \\dots, n)$) or\n$A = {2, 3, 4, 6}$.\nNote: The source [BaSo96] mentioned on the Erdős page makes it clear what\nquantifiers to use for \"where equality is achieved\". See Theorem 1.1 there.
\n\n TODO(firsching): Consider if we should have the other direction here as well or\nan iff statement.
"},"Erdos402.erdos_402.variants.szegedy_zaharescu_weak":{"url":"/FormalConjectures/ErdosProblems/«402»/#Erdos402___erdos_402___variants___szegedy_zaharescu_weak","anchor":"Erdos402___erdos_402___variants___szegedy_zaharescu_weak","docHtml":"\n Proved for all sufficiently large sets (including the sharper version which\ncharacterises the case of equality) independently by Szegedy [Sz86] and\nZaharescu [Za87]. The following is taken from [Sz86].
\n\n There exists an effectively computable $n_0$ with the following properties:\n(i) if $n \\ge n_0$ and $a_1, a_2, \\dots, a_n$ are distinct natural numbers then\n$\\max_{i, j} \\frac{a_i}{(a_i, a_j)} \\ge n$.\n(ii) If equality holds then the system ${a_1, a_2, \\dots, a_n}$ is either of the\ntype ${k, 2k, \\dots, nk}$ or of the type\n$\\left{\\frac{k}{1}, \\frac{k}{2}, \\dots, \\frac{k}{n}\\right}$.
"},"Erdos913.erdos_913":{"url":"/FormalConjectures/ErdosProblems/«913»/#Erdos913___erdos_913","anchor":"Erdos913___erdos_913","docHtml":"\n Are there infinitely many $n$ such that if\n$$\nn(n + 1) = \\prod_i p_i^{k_i}\n$$\nis the factorisation into distinct primes then all exponents $k_i$ are distinct?
"},"Erdos913.erdos_913.variants.infinite_many_8p_sq_add_one_primes":{"url":"/FormalConjectures/ErdosProblems/«913»/#Erdos913___erdos_913___variants___infinite_many_8p_sq_add_one_primes","anchor":"Erdos913___erdos_913___variants___infinite_many_8p_sq_add_one_primes","docHtml":"\n It is likely that there are infinitely many primes $p$ such that $8p^2 - 1$ is also prime.
"},"Erdos913.erdos_913.variants.conditional":{"url":"/FormalConjectures/ErdosProblems/«913»/#Erdos913___erdos_913___variants___conditional","anchor":"Erdos913___erdos_913___variants___conditional","docHtml":"\n If there are infinitely many primes $p$ such that $8p^2 - 1$ is prime, then this is true.
"},"Erdos283.Condition":{"url":"/FormalConjectures/ErdosProblems/«283»/#Erdos283___Condition","anchor":"Erdos283___Condition","docHtml":"\n Given a polynomial p, the predicate that if the leading coefficient is positive and\nthere exists no $d≥2$ with $d ∣ p(n)$ for all $n≥1$, then for all sufficiently large $m$,\nthere exist integers $1≤n_1<\\dots < n_k$ such that $$1=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k}$$\nand $$m=p(n_1)+\\cdots+p(n_k)$$?
\n Let $p\\colon \\mathbb{Z} \\rightarrow \\mathbb{Z}$ be a polynomial whose leading coefficient is\npositive and such that there exists no $d≥2$ with $d ∣ p(n)$ for all $n≥1$. Is it true that,\nfor all sufficiently large $m$, there exist integers $1≤n_1<\\dots < n_k$ such that\n$$1=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k}$$\nand\n$$m=p(n_1)+\\cdots+p(n_k)$$?
"},"Erdos283.erdos_283.variants.graham":{"url":"/FormalConjectures/ErdosProblems/«283»/#Erdos283___erdos_283___variants___graham","anchor":"Erdos283___erdos_283___variants___graham","docHtml":"\n Graham [Gr63] has proved this when $p(x)=x$.
"},"Erdos470.PrimitiveWeird":{"url":"/FormalConjectures/ErdosProblems/«470»/#Erdos470___PrimitiveWeird","anchor":"Erdos470___PrimitiveWeird","docHtml":"\n Primitive weird numbers are weird numbers such that no proper divisor of $n$ are weird.
"},"Erdos470.AbundancyIndex":{"url":"/FormalConjectures/ErdosProblems/«470»/#Erdos470___AbundancyIndex","anchor":"Erdos470___AbundancyIndex","docHtml":"\n The abundancy index is the sum of the divisors of $n$ divided by $n$.
"},"Erdos470.erdos_470.parts.i":{"url":"/FormalConjectures/ErdosProblems/«470»/#Erdos470___erdos_470___parts___i","anchor":"Erdos470___erdos_470___parts___i","docHtml":"\n Are there any odd weird numbers?
"},"Erdos470.erdos_470.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«470»/#Erdos470___erdos_470___parts___ii","anchor":"Erdos470___erdos_470___parts___ii","docHtml":"\n Are there infinitely many primitive weird numbers?
"},"Erdos470.erdos_470.variants.weird_pos_density":{"url":"/FormalConjectures/ErdosProblems/«470»/#Erdos470___erdos_470___variants___weird_pos_density","anchor":"Erdos470___erdos_470___variants___weird_pos_density","docHtml":"\n Benkoski and Erdős BeEr74 proved\nthat the set of weird numbers has positive density.
"},"Erdos470.erdos_470.variants.smallest_weird_eq_70":{"url":"/FormalConjectures/ErdosProblems/«470»/#Erdos470___erdos_470___variants___smallest_weird_eq_70","anchor":"Erdos470___erdos_470___variants___smallest_weird_eq_70","docHtml":"\n The smallest weird number is 70.
"},"Erdos470.erdos_470.variants.prime_gap_imp_inf_prim_weird":{"url":"/FormalConjectures/ErdosProblems/«470»/#Erdos470___erdos_470___variants___prime_gap_imp_inf_prim_weird","anchor":"Erdos470___erdos_470___variants___prime_gap_imp_inf_prim_weird","docHtml":"\n Melfi Me15 has proved that there\nare infinitely many primitive weird numbers, conditional on the fact that\n$p_{n+1} - p_n < \\frac{1}{10} \\sqrt{p_n}$ for all large $n$, which in turn would follow from\nwell-known conjectures concerning prime gaps.
"},"Erdos470.erdos_470.variants.odd_weird_10_pow_21":{"url":"/FormalConjectures/ErdosProblems/«470»/#Erdos470___erdos_470___variants___odd_weird_10_pow_21","anchor":"Erdos470___erdos_470___variants___odd_weird_10_pow_21","docHtml":"\n Fang Fa22 has shown there are no odd weird numbers below $10^{21}$.
"},"Erdos470.erdos_470.variants.odd_weird_prime_div":{"url":"/FormalConjectures/ErdosProblems/«470»/#Erdos470___erdos_470___variants___odd_weird_prime_div","anchor":"Erdos470___erdos_470___variants___odd_weird_prime_div","docHtml":"\n Liddy and Riedl LiRi18 have shown\nthat an odd weird number must have at least 6 prime divisors.
"},"Erdos470.erdos_470.variants.abundancy_index":{"url":"/FormalConjectures/ErdosProblems/«470»/#Erdos470___erdos_470___variants___abundancy_index","anchor":"Erdos470___erdos_470___variants___abundancy_index","docHtml":"\n If there are no odd weird numbers then every weird number has abundancy index < 4.
"},"Erdos249.erdos_249":{"url":"/FormalConjectures/ErdosProblems/«249»/#Erdos249___erdos_249","anchor":"Erdos249___erdos_249","docHtml":"\n Is\n$$\\sum_{n} \\frac{\\phi(n)}{2^n}$$\nirrational? Here $\\phi$ is the Euler totient function.
"},"Erdos1148.Erdos1148Prop":{"url":"/FormalConjectures/ErdosProblems/«1148»/#Erdos1148___Erdos1148Prop","anchor":"Erdos1148___Erdos1148Prop","docHtml":"\n A natural number $n$ which can be written as $n$ if $n = x^2 + y^2 - z^2$ with $\\max(x^2, y^2, z^2)\n\\leq n$.
"},"Erdos1148.erdos_1148":{"url":"/FormalConjectures/ErdosProblems/«1148»/#Erdos1148___erdos_1148","anchor":"Erdos1148___erdos_1148","docHtml":"\n Can every large integer $n$ be written as $n=x^2+y^2-z^2$ with $\\max(x^2,y^2,z^2)\\leq n$?
\n\n This was proved affirmatively by Chojecki [Ch26], using a Duke-type equidistribution theorem.\nA Lean formalisation of the reduction (conditional on a Duke-type equidistribution theorem) exists;\nsee the forum discussion.
"},"Erdos1148.erdos_1148.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«1148»/#Erdos1148___erdos_1148___variants___lower_bound","anchor":"Erdos1148___erdos_1148___variants___lower_bound","docHtml":"\n The integer $6563$ cannot be written as $x^2 + y^2 - z^2$ with $\\max(x^2, y^2, z^2) \\leq 6563$.
"},"Erdos1148.erdos_1148_weaker_prop":{"url":"/FormalConjectures/ErdosProblems/«1148»/#Erdos1148___erdos_1148_weaker_prop","anchor":"Erdos1148___erdos_1148_weaker_prop","docHtml":"\n The weaker property: $n = x^2 + y^2 - z^2$ such that $\\max(x^2, y^2, z^2) \\leq n + 2\\sqrt{n}$.
"},"Erdos1148.erdos_1148.variants.weaker":{"url":"/FormalConjectures/ErdosProblems/«1148»/#Erdos1148___erdos_1148___variants___weaker","anchor":"Erdos1148___erdos_1148___variants___weaker","docHtml":"\n [Va99] reports this is 'obvious' if we replace $\\leq n$ with $\\leq n+2\\sqrt{n}$.
"},"Erdos825.erdos_825":{"url":"/FormalConjectures/ErdosProblems/«825»/#Erdos825___erdos_825","anchor":"Erdos825___erdos_825","docHtml":"\n Is there an absolute constant $C > 0$ such that every integer $n$ with\n$\\sigma(n) > Cn$ is the distinct sum of proper divisors of $n$?
\n\n This has been solved in the affirmative by Larsen - in fact, for any $\\epsilon>0$ there exists $L$\nsuch that if $n$ has only prime divisors $>L$ and $\\sigma(n)>(2+\\epsilon)n$ then $n$ is the distinct\nsum of proper divisors of $n$.
"},"Erdos825.erdos_825.variants.necessary_cond":{"url":"/FormalConjectures/ErdosProblems/«825»/#Erdos825___erdos_825___variants___necessary_cond","anchor":"Erdos825___erdos_825___variants___necessary_cond","docHtml":"\n Show that if the constant $C > 0$ is such that every integer $n$ with\n$\\sigma(n) > Cn$ is the distinct sum of proper divisors of $n$, then we\nmust have $C > 2$.
"},"Erdos168.NonTernary":{"url":"/FormalConjectures/ErdosProblems/«168»/#Erdos168___NonTernary","anchor":"Erdos168___NonTernary","docHtml":"\n Say a finite set of natural numbers is n, 2n, 3n.
\nIntervalNonTernarySets N is the (fin)set of non ternary subsets of {1,...,N}.\nThe advantage of defining it as below is that some proofs (e.g. that of F 3 = 2) become rfl.
\nF N is the size of the largest non ternary subset of {1,...,N}.
\n Sanity check: elements of IntervalNonTernarySets N are precisely non ternary subsets of\n{1,...,N}
\n Sanity check: if S is a maximal non ternary subset of {1,..., N} then F N is given by the\ncardinality of S
\n What is the limit $F(N)/N$ as $N \\to \\infty$?
"},"Erdos168.erdos_168.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«168»/#Erdos168___erdos_168___parts___ii","anchor":"Erdos168___erdos_168___parts___ii","docHtml":"\n Is the limit $F(N)/N$ as $N \\to \\infty$ irrational?
"},"Erdos168.erdos_168.variants.limit_exists":{"url":"/FormalConjectures/ErdosProblems/«168»/#Erdos168___erdos_168___variants___limit_exists","anchor":"Erdos168___erdos_168___variants___limit_exists","docHtml":"\n The limit $F(N)/N$ as $N \\to \\infty$ exists. (proved by Graham, Spencer, and Witsenhausen)
"},"Erdos1108.FactorialSums":{"url":"/FormalConjectures/ErdosProblems/«1108»/#Erdos1108___FactorialSums","anchor":"Erdos1108___FactorialSums","docHtml":"\n The set $A = \\left{ \\sum_{n\\in S}n! : S\\subset \\mathbb{N}\\text{ finite}\\right}$ of all finite\nsums of distinct factorials.
"},"Erdos1108.IsPowerful":{"url":"/FormalConjectures/ErdosProblems/«1108»/#Erdos1108___IsPowerful","anchor":"Erdos1108___IsPowerful","docHtml":"\n A number is powerful if each prime factor appears with exponent at least 2.
"},"Erdos1108.erdos_1108.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1108»/#Erdos1108___erdos_1108___parts___i","anchor":"Erdos1108___erdos_1108___parts___i","docHtml":"\n For each $k \\geq 2$, does the set $A = \\left{ \\sum_{n\\in S}n! : S\\subset \\mathbb{N}\\text{ finite}\\right}$ of all finite sums of distinct factorials contain only finitely many $k$-th powers?
"},"Erdos1108.erdos_1108.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«1108»/#Erdos1108___erdos_1108___parts___ii","anchor":"Erdos1108___erdos_1108___parts___ii","docHtml":"\n Does the set $A = \\left{ \\sum_{n\\in S}n! : S\\subset \\mathbb{N}\\text{ finite}\\right}$ of all finite sums of distinct factorials contain only finitely many powerful numbers?
"},"Erdos505.erdos_505.test_dim_one":{"url":"/FormalConjectures/ErdosProblems/«505»/#Erdos505___erdos_505___test_dim_one","anchor":"Erdos505___erdos_505___test_dim_one","docHtml":"\n
\nBorsuk's conjecture (1933): Is every bounded set of diameter 1 in $\\mathbb{R}^n$\nthe union of at most $n + 1$ sets of diameter strictly less than 1?
\n\n Erdős [Er44] suspected this is false for sufficiently large $n$. Confirmed\nby Kahn–Kalai [KK93], who disproved the conjecture for $n \\geq 2015$.\nThe current best is $n \\geq 64$ (Jenrich–Brouwer, 2014).
\n\n The conjecture is true for $n \\leq 3$ (Eggleston [Eg55] for $n = 3$).
\n\n [Bo33] Borsuk, K. (1933).
\n [Er44] Erdős, P. (1944). Remarks on a conjecture of Borsuk.
\n\n [Eg55] Eggleston, H. G. (1955).
\n [KK93] Kahn, J., Kalai, G. (1993).
\n Lean 4 code in this file was drafted with assistance from Claude (Anthropic).\nThe mathematical content and references are the author's own work.
"},"Erdos505.erdos_505":{"url":"/FormalConjectures/ErdosProblems/«505»/#Erdos505___erdos_505","anchor":"Erdos505___erdos_505","docHtml":"\nErdős Problem 505 (disproved). Borsuk's conjecture is false for\nsufficiently large $n$: there exists a dimension $n$ and a bounded set\n$S \\subseteq \\mathbb{R}^n$ with positive diameter such that $S$ cannot be\ncovered by $n + 1$ subsets each of diameter strictly less than $\\operatorname{diam}(S)$.
\n\n Erdős [Er44] suspected this. Disproved by Kahn–Kalai [KK93] for\n$n \\geq 2015$. Currently known to be false for $n \\geq 64$.\nA formal proof was formalised by Boris Alexeev using Aristotle.
"},"Erdos505.erdos_505.small_dim":{"url":"/FormalConjectures/ErdosProblems/«505»/#Erdos505___erdos_505___small_dim","anchor":"Erdos505___erdos_505___small_dim","docHtml":"\nBorsuk's conjecture, small dimensions (open / true for $n \\leq 3$).\nEvery bounded set $S \\subseteq \\mathbb{R}^n$ with $n \\leq 3$ can be\ncovered by $n + 1$ subsets each of strictly smaller diameter.
\n\n Trivial for $n \\leq 2$; proved for $n = 3$ by Eggleston [Eg55].
"},"Erdos961.Erdos961Prop":{"url":"/FormalConjectures/ErdosProblems/«961»/#Erdos961___Erdos961Prop","anchor":"Erdos961___Erdos961Prop","docHtml":"\n
\n [Ju74] Jutila, Matti, On numbers with a large prime factor. {II}. J. Indian Math. Soc. (N.S.) (1974), 125--130.
\n\nRaSh73 Ramachandra, K. and Shorey, T. N., On gaps between numbers with a large prime factor. Acta Arith. (1973), 99--111.
\n\n Sylvester and Schur [Er34] proved that every set of $k$ consecutive integers greater than $k$\ncontains an integer divisible by a prime greater than $k$, i.e. not $(k+1)$-smooth.
"},"Erdos961.erdos_961.variants.sylvester_schur_1_1":{"url":"/FormalConjectures/ErdosProblems/«961»/#Erdos961___erdos_961___variants___sylvester_schur_1_1","anchor":"Erdos961___erdos_961___variants___sylvester_schur_1_1"},"Erdos961.erdos_961.variants.well_defined":{"url":"/FormalConjectures/ErdosProblems/«961»/#Erdos961___erdos_961___variants___well_defined","anchor":"Erdos961___erdos_961___variants___well_defined","docHtml":"\n There exists $n$ such that Erdos961Prop k n holds.
\n For $k$, let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains\nan integer divisible by a prime $>k$, i.e. not $(k+1)$-smooth.
"},"Erdos961.erdos_961":{"url":"/FormalConjectures/ErdosProblems/«961»/#Erdos961___erdos_961","anchor":"Erdos961___erdos_961","docHtml":"\n It is conjectured that $f(k) \\ll (\\log k)^O(1)$.
"},"Erdos961.erdos_961.variants.erdos_upper_bound":{"url":"/FormalConjectures/ErdosProblems/«961»/#Erdos961___erdos_961___variants___erdos_upper_bound","anchor":"Erdos961___erdos_961___variants___erdos_upper_bound","docHtml":"\n Erdos [Er55d] proved $f(k) < 3 \\frac{k}{\\log k}$ for sufficiently large $k$.
"},"Erdos961.erdos_961.variants.jutila_ramachandra_shorey_upper_bound":{"url":"/FormalConjectures/ErdosProblems/«961»/#Erdos961___erdos_961___variants___jutila_ramachandra_shorey_upper_bound","anchor":"Erdos961___erdos_961___variants___jutila_ramachandra_shorey_upper_bound","docHtml":"\n Jutila [Ju74], and Ramachandra--Shorey [RaSh73] proved a stronger upper bound\n$f(k) \\ll \\frac{\\log \\log \\log k}{\\log \\log k} \\frac{k}{\\log k}$.
"},"Erdos1038.erdos_1038.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1038»/#Erdos1038___erdos_1038___parts___i","anchor":"Erdos1038___erdos_1038___parts___i","docHtml":"\n What is the infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such\nthat all of its roots are real and contained in [-1,1]?
\n The supremum of |{x ∈ ℝ : |f x| < 1}| over all monic polynomials f such that\nall of its roots are real and contained in [-1,1] is 2 * 2 ^ (1 / 2). This is proved in\n[Tao25].
\n The infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that\nall of its roots are real and contained in [-1,1] is < 1.835.
\n The infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that\nall of its roots are real and contained in [-1,1] is ≥ 2 ^ (4 / 3) - 1.
\n
\n Is it true that, for any $C > 0$, there infinitely many $n$ such that:\n$$\np_{n + 1} - p_n > C \\frac{\\log\\log n\\log\\log\\log\\log n}{(\\log\\log\\log n) ^ 2}\\log n\n$$
"},"Erdos4.erdos_4.variants.rankin":{"url":"/FormalConjectures/ErdosProblems/«4»/#Erdos4___erdos_4___variants___rankin","anchor":"Erdos4___erdos_4___variants___rankin","docHtml":"\n Rankin's theorem: there exists a positive constant $C$ such that Erdos4For C holds.
\n Let $1\\leq m\\leq k$ and $f_{k,m}(x)$ denote the number of integers $\\leq x$ which are the sum of\n$m$ many nonnegative $k$th powers.
"},"Erdos323.erdos_323.parts.i":{"url":"/FormalConjectures/ErdosProblems/«323»/#Erdos323___erdos_323___parts___i","anchor":"Erdos323___erdos_323___parts___i","docHtml":"\n Is it true that $f_{k,k}(x) \\gg_\\epsilon x^{1-\\epsilon}$ for all $\\epsilon>0$?
\n\n This would have significant applications to Waring's problem. Erdős and Graham describe this as\n'unattackable by the methods at our disposal'.
"},"Erdos323.erdos_323.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«323»/#Erdos323___erdos_323___parts___ii","anchor":"Erdos323___erdos_323___parts___ii","docHtml":"\n Is it true that if $m < k$ then $f_{k,m}(x) \\gg x^{m/k}$ for sufficiently large $x$?
"},"Erdos323.erdos_323.variants.k_eq_2":{"url":"/FormalConjectures/ErdosProblems/«323»/#Erdos323___erdos_323___variants___k_eq_2","anchor":"Erdos323___erdos_323___variants___k_eq_2","docHtml":"\n The case $k=2$ was resolved by Landau, who showed $f_{2,2}(x) \\sim \\frac{cx}{\\sqrt{\\log x}}$ for\nsome constant $c>0$.
"},"Erdos323.erdos_323.variants.k_gt_2":{"url":"/FormalConjectures/ErdosProblems/«323»/#Erdos323___erdos_323___variants___k_gt_2","anchor":"Erdos323___erdos_323___variants___k_gt_2","docHtml":"\n For $k>2$ it is not known if $f_{k,k}(x)=o(x)$.
"},"Erdos371.erdos_371":{"url":"/FormalConjectures/ErdosProblems/«371»/#Erdos371___erdos_371","anchor":"Erdos371___erdos_371","docHtml":"\n Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$\nwith $P(n+1) > P(n)$ has density $\\frac{1}{2}$.
"},"Erdos1049.erdos_1049":{"url":"/FormalConjectures/ErdosProblems/«1049»/#Erdos1049___erdos_1049","anchor":"Erdos1049___erdos_1049","docHtml":"\n Let $t>1$ be a rational number. Is\n$\\sum_{n=1}^\\infty\\frac{1}{t^n-1}=\\sum_{n=1}^\\infty \\frac{\\tau(n)}{t^n}$ irrational, where\n$\\tau(n)$ counts the divisors of $n$?
\n\n A conjecture of Chowla.
"},"Erdos1049.erdos_1049.variants.geq_2_integer":{"url":"/FormalConjectures/ErdosProblems/«1049»/#Erdos1049___erdos_1049___variants___geq_2_integer","anchor":"Erdos1049___erdos_1049___variants___geq_2_integer","docHtml":"\n Erdős [Er48] proved that this is true if $t\\geq 2$ is an integer.
"},"Erdos1049.lambert_series_eq_num_divisor_sum":{"url":"/FormalConjectures/ErdosProblems/«1049»/#Erdos1049___lambert_series_eq_num_divisor_sum","anchor":"Erdos1049___lambert_series_eq_num_divisor_sum","docHtml":"\n The classical Lambert series identity: $\\sum_{n=1}^\\infty \\frac{1}{t^n - 1} =\n\\sum_{n=1}^\\infty \\frac{\\tau(n)}{t^n}$, where $\\tau(n)$ counts the divisors of $n$.
"},"Erdos849.erdos_849":{"url":"/FormalConjectures/ErdosProblems/«849»/#Erdos849___erdos_849","anchor":"Erdos849___erdos_849","docHtml":"\n Is it true that, for every integer $t\\geq1$, there is some integer $a$ such that ${n \\choose k} = a$\nwith $1\\leq k \\le \\frac{n}{2}$ has exactly $t$ solutions?
"},"Erdos613.erdos_613":{"url":"/FormalConjectures/ErdosProblems/«613»/#Erdos613___erdos_613","anchor":"Erdos613___erdos_613","docHtml":"\nErdős Problem 613:\nLet $n \\geq 3$ and $G$ be a graph with $\\binom{2n+1}{2} - \\binom{n}{2} - 1$ edges.\nMust $G$ be the union of a bipartite graph and a graph with maximum degree less than $n$?
"},"Erdos699.sylvester_schur":{"url":"/FormalConjectures/ErdosProblems/«699»/#Erdos699___sylvester_schur","anchor":"Erdos699___sylvester_schur","docHtml":"\n Sylvester and Schur: for $1 \\le i \\le n/2$ there is a prime $p > i$ dividing n.choose i.
\nErdős Problem 699. Is it true that for every $1 \\le i < j \\le n / 2$ there exists a prime\n$p \\ge i$ with $p \\mid \\gcd\\big(\\binom{n}{i}, \\binom{n}{j}\\big)$?
"},"Erdos699.erdos_szekeres_strengthening":{"url":"/FormalConjectures/ErdosProblems/«699»/#Erdos699___erdos_szekeres_strengthening","anchor":"Erdos699___erdos_szekeres_strengthening","docHtml":"\n Erdős and Szekeres conjectured that, apart from a finite exceptional set of triples (n, i, j),\none can always take p > i in the prime divisor statement.
\n How many iterations of $n\\mapsto\\phi(n) + 1$ are needed before a prime is reached?
"},"Erdos409.erdos_409.variants.termination":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___variants___termination","anchor":"Erdos409___erdos_409___variants___termination","docHtml":"\n If $n > 0$, then the iteration $n\\mapsto\\phi(n) + 1$ necessarily\nreaches a prime.
"},"Erdos409.erdos_409.parts.i.isTheta":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___parts___i___isTheta","anchor":"Erdos409___erdos_409___parts___i___isTheta","docHtml":"\n Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\phi(n) + 1$ before a prime\nis reached. What is $\\Theta(c(n))$?
"},"Erdos409.erdos_409.parts.i.isBigO":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___parts___i___isBigO","anchor":"Erdos409___erdos_409___parts___i___isBigO","docHtml":"\n Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\phi(n) + 1$ before a prime\nis reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?
"},"Erdos409.erdos_409.parts.i.isLittleO":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___parts___i___isLittleO","anchor":"Erdos409___erdos_409___parts___i___isLittleO","docHtml":"\n Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\phi(n) + 1$ before a prime\nis reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?
"},"Erdos409.erdos_409.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___parts___ii","anchor":"Erdos409___erdos_409___parts___ii","docHtml":"\n Can infinitely many $n$ reach the same prime under the iteration $n\\mapsto\\phi(n) + 1$?
"},"Erdos409.erdos_409.parts.iii":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___parts___iii","anchor":"Erdos409___erdos_409___parts___iii","docHtml":"\n What is the density of $n$ which reach any fixed prime under the iteration $n\\mapsto\\phi(n) + 1$?
"},"Erdos409.erdos_409.variants.sigma":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___variants___sigma","anchor":"Erdos409___erdos_409___variants___sigma","docHtml":"\n How many iterations of $n\\mapsto\\sigma(n) - 1$ are needed before a prime is reached?
"},"Erdos409.erdos_409.variants.sigma_termination":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___variants___sigma_termination","anchor":"Erdos409___erdos_409___variants___sigma_termination","docHtml":"\n If $n > 1$ then the iteration $n\\mapsto\\sigma(n) - 1$ necessarily reaches a prime.\nNote: this is open — it is not clear that the σ iteration always terminates,\nsince it is non-decreasing (unlike the φ iteration which is strictly decreasing).
"},"Erdos409.erdos_409.variants.sigma_isTheta":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___variants___sigma_isTheta","anchor":"Erdos409___erdos_409___variants___sigma_isTheta","docHtml":"\n Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\sigma(n) - 1$ before a prime\nis reached. What is $\\Theta(c(n))$?
"},"Erdos409.erdos_409.variants.sigma_isBigO":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___variants___sigma_isBigO","anchor":"Erdos409___erdos_409___variants___sigma_isBigO","docHtml":"\n Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\sigma(n) - 1$ before a prime\nis reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?
"},"Erdos409.erdos_409.variants.sigma_isLittleO":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___variants___sigma_isLittleO","anchor":"Erdos409___erdos_409___variants___sigma_isLittleO","docHtml":"\n Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\sigma(n) - 1$ before a prime\nis reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?
"},"Erdos409.erdos_409.variants.sigma_prime_termination":{"url":"/FormalConjectures/ErdosProblems/«409»/#Erdos409___erdos_409___variants___sigma_prime_termination","anchor":"Erdos409___erdos_409___variants___sigma_prime_termination","docHtml":"\n Is it true that iterates of $n\\mapsto\\sigma(n) - 1$ always reach a prime?
"},"Erdos1004.IsDistinctTotientRun":{"url":"/FormalConjectures/ErdosProblems/«1004»/#Erdos1004___IsDistinctTotientRun","anchor":"Erdos1004___IsDistinctTotientRun","docHtml":"\nIsDistinctTotientRun n K means that the values φ(n+1), φ(n+2), ..., φ(n+K) are all distinct.
\n For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that\nthe values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c.\nThis is an open problem.
"},"Erdos1004.erdos_1004.variants.le_of_isDistinctTotientRun":{"url":"/FormalConjectures/ErdosProblems/«1004»/#Erdos1004___erdos_1004___variants___le_of_isDistinctTotientRun","anchor":"Erdos1004___erdos_1004___variants___le_of_isDistinctTotientRun","docHtml":"\n Erdős, Pomerance, and Sárközy [EPS87] proved that if φ(n+k) are all distinct for 1 ≤ k ≤ K then\nK ≤ n / exp(c (log n)^{1/3}) for some constant c > 0.\nHere we state the existence of such a constant c.
"},"Erdos723.erdos_723":{"url":"/FormalConjectures/ErdosProblems/«723»/#Erdos723___erdos_723","anchor":"Erdos723___erdos_723","docHtml":"\n If there is a finite projective plane of order $n$ then must $n$ be a prime power?
"},"Erdos723.erdos_723.variants.prime_power_is_projplane_order":{"url":"/FormalConjectures/ErdosProblems/«723»/#Erdos723___erdos_723___variants___prime_power_is_projplane_order","anchor":"Erdos723___erdos_723___variants___prime_power_is_projplane_order","docHtml":"\n These always exist if $n$ is a prime power.
"},"Erdos723.erdos_723.variants.leq_11":{"url":"/FormalConjectures/ErdosProblems/«723»/#Erdos723___erdos_723___variants___leq_11","anchor":"Erdos723___erdos_723___variants___leq_11","docHtml":"\n This conjecture has been proved for $n \\leq 11$.
"},"Erdos723.erdos_723.variants.eq_12":{"url":"/FormalConjectures/ErdosProblems/«723»/#Erdos723___erdos_723___variants___eq_12","anchor":"Erdos723___erdos_723___variants___eq_12","docHtml":"\n It is open whether there exists a projective plane of order 12.
"},"Erdos723.erdos_723.variants.bruck_ryser":{"url":"/FormalConjectures/ErdosProblems/«723»/#Erdos723___erdos_723___variants___bruck_ryser","anchor":"Erdos723___erdos_723___variants___bruck_ryser","docHtml":"\n Bruck and Ryser have proved that if $n \\equiv 1 (\\mod 4)$ or $n \\equiv 2 (\\mod 4)$ then $n$ must be\nthe sum of two squares.
"},"Erdos41.NtupleCondition":{"url":"/FormalConjectures/ErdosProblems/«41»/#Erdos41___NtupleCondition","anchor":"Erdos41___NtupleCondition","docHtml":"\n For a given set A, the n-tuple sums a₁ + ... + aₙ are all distinct for a₁, ..., aₙ in A\n(aside from the trivial coincidences).
\n Let A ⊆ ℕ be an infinite set such that the triple sums a + b + c are all distinct for\na, b, c in A (aside from the trivial coincidences). Is it true that\nliminf n → ∞ |A ∩ {1, …, N}| / N^(1/3) = 0?
\n Erdős proved the following pairwise version.\nLet A ⊆ ℕ be an infinite set such that the pairwise sums a + b are all distinct for a, b\nin A (aside from the trivial coincidences).\nIs it true that liminf n → ∞ |A ∩ {1, …, N}| / N^(1/2) = 0?
\n Does every graph with chromatic number $\\aleph_1$ contain a countable subgraph which is\ninfinitely connected?
"},"Erdos946.erdos_946":{"url":"/FormalConjectures/ErdosProblems/«946»/#Erdos946___erdos_946","anchor":"Erdos946___erdos_946","docHtml":"\n There are infinitely many $n$ such that $τ(n) = τ(n+1)$. Proved in [He84].\nHere τ is the divisor counting function, which is σ 0 in mathlib.
\n There are infinitely many $n$ such that $τ(n) = τ(n + 5040)$. Proved in [Sp81].
"},"Erdos946.erdos946Count":{"url":"/FormalConjectures/ErdosProblems/«946»/#Erdos946___erdos946Count","anchor":"Erdos946___erdos946Count","docHtml":"\n Number of $n \\le x$ with $τ(n) = τ(n+1)$.
"},"Erdos946.erdos_946.variants.heathbrown_lower_bound":{"url":"/FormalConjectures/ErdosProblems/«946»/#Erdos946___erdos_946___variants___heathbrown_lower_bound","anchor":"Erdos946___erdos_946___variants___heathbrown_lower_bound","docHtml":"\n The number of $n \\le x$ with $τ(n) = τ(n+1)$ is at least $x / (\\log x)^7$ for all sufficiently\nlarge $x$. Proved in [He84].
"},"Erdos946.erdos_946.variants.hildebrand_lower_bound":{"url":"/FormalConjectures/ErdosProblems/«946»/#Erdos946___erdos_946___variants___hildebrand_lower_bound","anchor":"Erdos946___erdos_946___variants___hildebrand_lower_bound","docHtml":"\n Improved lower bound in [Hi85]: $Ω(x / (\\log \\log x)^3)$.
"},"Erdos946.erdos_946.variants.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«946»/#Erdos946___erdos_946___variants___upper_bound","anchor":"Erdos946___erdos_946___variants___upper_bound","docHtml":"\n Upper bound in [EPS87]: $O(x / \\sqrt{\\log \\log x})$.
"},"Erdos851.TwoPowAddSet":{"url":"/FormalConjectures/ErdosProblems/«851»/#Erdos851___TwoPowAddSet","anchor":"Erdos851___TwoPowAddSet","docHtml":"\nTwoPowAddSet r is the set of integers of the form 2^k+n, where k ≥ 0 and n has at most r\nprime divisors.
\n The set of integers of the form 2^k+p (where p is prime) has positive lower density.
\n Formalisation note: here we also allow p = 1 since this simplifies the code and is equivalent\nto the original statement.
\n Let $\\epsilon > 0$. Is there some $r \\ll_\\epsilon 1$ such that the density of integers of the\nform $2^k+n$, where $k \\geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\\epsilon$?
\n\n This was proved affirmatively by Price and GPT-5.2 Pro [Pr26].
"},"Erdos513.ratio":{"url":"/FormalConjectures/ErdosProblems/«513»/#Erdos513___ratio","anchor":"Erdos513___ratio","docHtml":"\n
\n [ClHa64] Clunie, J. and Hayman, W. K., The maximum term of a power series. J. Analyse Math.\n(1964), 143-186.
\n\n Let f be a transcendental entire function. What is the greatest possible value of\nliminf (fun r : ℝ => ratio r f) atTop?
\n For all transcendental entire function f, liminf (fun r : ℝ => ratio r f) atTop ≤ 2 / π - c\nfor some c > 0. This is proved in [ClHa64].
\n For all transcendental entire function f, liminf (fun r : ℝ => ratio r f) atTop > 1 / 2.
\n The proposition that the modular product of a collection of consecutive interval equals $1$ modulo $p$,\nwhere intervals are defined by a function specifying the consecutive boundaries.
"},"Erdos1056.erdos_1056":{"url":"/FormalConjectures/ErdosProblems/«1056»/#Erdos1056___erdos_1056","anchor":"Erdos1056___erdos_1056","docHtml":"\n Let $k ≥ 2$. Does there exist a prime $p$ and consecutive intervals $I_0,\\dots,I_k$\nsuch that $\\prod\\limits_{n{\\in}I_i}n \\equiv 1 \\mod n$ for all $1 \\le i \\le k$?
"},"Erdos1056.erdos_1056.variants.k2":{"url":"/FormalConjectures/ErdosProblems/«1056»/#Erdos1056___erdos_1056___variants___k2","anchor":"Erdos1056___erdos_1056___variants___k2","docHtml":"\n This is problem A15 in Guy's collection [Gu04], where he reports that in a letter in 1979\nErdős observed that $3 * 4 \\equiv 5 * 6 * 7 \\equiv 1 \\mod 11$.
"},"Erdos1056.erdos_1056.variants.k3":{"url":"/FormalConjectures/ErdosProblems/«1056»/#Erdos1056___erdos_1056___variants___k3","anchor":"Erdos1056___erdos_1056___variants___k3","docHtml":"\n Makowski [Ma83] found, for $k=3$:\n$2 * 3 * 4 * 5 \\equiv 6 * 7 * 8 * 9 * 10 * 11 \\equiv 12 * 13 * 14 * 15 \\equiv 1 \\mod 17$.
"},"Erdos1056.erdos_1056.variants.noll_simmons":{"url":"/FormalConjectures/ErdosProblems/«1056»/#Erdos1056___erdos_1056___variants___noll_simmons","anchor":"Erdos1056___erdos_1056___variants___noll_simmons","docHtml":"\n Noll and Simmons asked, more generally, whether there are solutions to\n$q_1! \\equiv \\dots \\equiv q_k! \\mod p$ for arbitrarily large $k$ (with $q_1 < \\dots < q_k$).
"},"Erdos587.MaxNotSqSum":{"url":"/FormalConjectures/ErdosProblems/«587»/#Erdos587___MaxNotSqSum","anchor":"Erdos587___MaxNotSqSum","docHtml":"\nMaxNotSqSum N is the size of the largest subset A of\n{1,...,N} such that for all non-empty S ⊆ A, the sum\n∑ n ∈ S, n is not a square.
\n Nguyen and Vu proved that $|A| \\ll N^{1/3} (\\log N)^{O(1)}$.
"},"Erdos1203.F":{"url":"/FormalConjectures/ErdosProblems/«1203»/#Erdos1203___F","anchor":"Erdos1203___F","docHtml":"\n If $\\omega(n)$ counts the number of distinct prime divisors of $n$ then let\n$F(n)=\\max_k \\omega(n+k)\\frac{\\log\\log k}{\\log k}.$
"},"Erdos1203.erdos_1203":{"url":"/FormalConjectures/ErdosProblems/«1203»/#Erdos1203___erdos_1203","anchor":"Erdos1203___erdos_1203","docHtml":"\n Prove that $F(n)\\to \\infty$ as $n\\to \\infty$.
"},"Erdos1203.erdos_1203.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«1203»/#Erdos1203___erdos_1203___variants___lower_bound","anchor":"Erdos1203___erdos_1203___variants___lower_bound","docHtml":"\n It is easy to prove that $F(n)\\geq 1-o(1)$.
"},"Erdos889.v":{"url":"/FormalConjectures/ErdosProblems/«889»/#Erdos889___v","anchor":"Erdos889___v","docHtml":"\n $v(n,k)$ counts the prime factors of $n+k$ which do not divide $n+i$\nfor all $0 \\le i < k$.
"},"Erdos889.v₀":{"url":"/FormalConjectures/ErdosProblems/«889»/#Erdos889___v___","anchor":"Erdos889___v___","docHtml":"\n $v_0(n)$ is the supremum of $v(n,k)$ for all $k \\ge 0$.
"},"Erdos889.erdos_889":{"url":"/FormalConjectures/ErdosProblems/«889»/#Erdos889___erdos_889","anchor":"Erdos889___erdos_889","docHtml":"\n Let $v(n,k)$ count the prime factors of $n+k$ which\ndo not divide $n+i$ for $0\\leq i < k$. Is it true that\n$v_0(n)=\\max_{k\\geq 0}v(n,k)\\to \\infty$ as $n\\to \\infty$?
"},"Erdos889.erdos_889.variants.v0_gt_1":{"url":"/FormalConjectures/ErdosProblems/«889»/#Erdos889___erdos_889___variants___v0_gt_1","anchor":"Erdos889___erdos_889___variants___v0_gt_1","docHtml":"\n $v_0(n) > 1$ for all $n$ except $n$ = 0, 1, 2, 3, 4, 7, 8, 16
\n\n [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.
"},"Erdos889.v_l":{"url":"/FormalConjectures/ErdosProblems/«889»/#Erdos889___v_l","anchor":"Erdos889___v_l","docHtml":"\n $v_l(n)$ is the supremum of $v(n,k)$ for all $k \\ge l$
"},"Erdos889.erdos_889.variants.general":{"url":"/FormalConjectures/ErdosProblems/«889»/#Erdos889___erdos_889___variants___general","anchor":"Erdos889___erdos_889___variants___general","docHtml":"\n Let $v_l(n) = \\max_{k\\geq l} v(n,k)$. For every fixed $l$,\n$v_l(n) \\to \\infty$ as $n \\to \\infty$
\n\n [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.
"},"Erdos889.erdos_889.variants.v1_eq_1_finite":{"url":"/FormalConjectures/ErdosProblems/«889»/#Erdos889___erdos_889___variants___v1_eq_1_finite","anchor":"Erdos889___erdos_889___variants___v1_eq_1_finite","docHtml":"\n Does $v_1(n) = 1$ have finite solutions?
\n\n [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.
"},"Erdos889.V":{"url":"/FormalConjectures/ErdosProblems/«889»/#Erdos889___V","anchor":"Erdos889___V","docHtml":"\n $V(n,k)$ is the number of primes $p$ such that\n$p^\\alpha$ exactly divides $n+k$ and\nfor all $0 \\le i < k$, $p^\\alpha$ does not divide $n+i$,\nwhere $\\alpha$ is the multiplicity of $p$ in the factorization of $n+k$.
"},"Erdos889.V_l":{"url":"/FormalConjectures/ErdosProblems/«889»/#Erdos889___V_l","anchor":"Erdos889___V_l","docHtml":"\n $V_l(n)$ is the supremum of $V(n,k)$ for all $k \\ge l$
"},"Erdos889.erdos_889.variants.V1_eq_1_finite":{"url":"/FormalConjectures/ErdosProblems/«889»/#Erdos889___erdos_889___variants___V1_eq_1_finite","anchor":"Erdos889___erdos_889___variants___V1_eq_1_finite","docHtml":"\n Does $V_1(n) = 1$ have finite solutions?
\n\n This is a modification of erdos_889.variants.v1_eq_1_finite,\nwhich might make it more amenable to attack according to [ErSe67].
\n [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.
"},"Erdos153.f":{"url":"/FormalConjectures/ErdosProblems/«153»/#Erdos153___f","anchor":"Erdos153___f","docHtml":"\n Define $f(n)$ to be the minimum of\n$\\frac{1}{t}\\sum_{1\\leq i<t}(s_{i+1}-s_i)^2$ as $A$ ranges over all Sidon sets of size $n$, where\n$A+A={s_1<\\cdots<s_t}$.
"},"Erdos153.erdos_153":{"url":"/FormalConjectures/ErdosProblems/«153»/#Erdos153___erdos_153","anchor":"Erdos153___erdos_153","docHtml":"\n Let $A$ be a finite Sidon set and $A+A={s_1<\\cdots<s_t}$. Is it true that\n$$\\frac{1}{t}\\sum_{1\\leq i<t}(s_{i+1}-s_i)^2 \\to \\infty$$\nas $\\lvert A\\rvert\\to \\infty$?
"},"Erdos98.h":{"url":"/FormalConjectures/ErdosProblems/«98»/#Erdos98___h","anchor":"Erdos98___h","docHtml":"\n $h(n)$ is the minimum number of distinct distances determined by any\n$n$-point set in $\\mathbb{R}^2$ in general position (no three collinear, no four\ncocyclic).
"},"Erdos98.erdos_98":{"url":"/FormalConjectures/ErdosProblems/«98»/#Erdos98___erdos_98","anchor":"Erdos98___erdos_98","docHtml":"\n Let $h(n)$ be such that any $n$ points in $\\mathbb{R}^2$, with no three on a line\nand no four on a circle, determine at least $h(n)$ distinct distances. Does\n$h(n)/n\\to \\infty$?
"},"Erdos98.erdos_98.variants.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«98»/#Erdos98___erdos_98___variants___upper_bound","anchor":"Erdos98___erdos_98___variants___upper_bound","docHtml":"\n Erdős could not even prove $h(n)\\geq n$. Pach has shown $h(n) < n^{\\log_2 3}$.\nErdős, Füredi, and Pach [EFPR93] have improved this to\n$h(n) < n\\exp(c\\sqrt{\\log n})$ for some constant $c>0$.
"},"Erdos705.erdos_705":{"url":"/FormalConjectures/ErdosProblems/«705»/#Erdos705___erdos_705","anchor":"Erdos705___erdos_705","docHtml":"\n Let $G$ be a finite unit distance graph in $\\mamthbb{R}^2$.\nIs there some $k$ such that if $G$ has girth $≥ k$, then $\\chi(G) ≤ 3$?
\n\n The general case was solved by O'Donnell [OD99], who constructed finite unit distance graphs with\nchromatic number $4$ and arbitrarily large girth.
"},"Erdos267.erdos_267":{"url":"/FormalConjectures/ErdosProblems/«267»/#Erdos267___erdos_267","anchor":"Erdos267___erdos_267","docHtml":"\n Let $F_1=F_2=1$ and $F_{n+1} = F_n + F_{n-1}$ be the Fibonacci sequence.\nLet $n_1 < n_2 < \\dots$ be an infinite sequence with $\\frac{n_{k+1}}{n_k} \\ge c > 1$. Must\n$\\sum_k \\frac 1 {F_{n_k}}$ be irrational?
"},"Erdos267.erdos_267.variants.generalisation_ratio_limit_to_infinity":{"url":"/FormalConjectures/ErdosProblems/«267»/#Erdos267___erdos_267___variants___generalisation_ratio_limit_to_infinity","anchor":"Erdos267___erdos_267___variants___generalisation_ratio_limit_to_infinity","docHtml":"\n Let $F_1=F_2=1$ and $F_{n+1} = F_n + F_{n-1}$ be the Fibonacci sequence.\nLet $n_1 < n_2 < \\dots$ be an infinite sequence with $\\frac {n_k}{k} \\to \\infty$. Must\n$\\sum_k \\frac 1 {F_{n_k}}$ be irrational?
"},"Erdos267.erdos_267.variants.specialization_pow_two":{"url":"/FormalConjectures/ErdosProblems/«267»/#Erdos267___erdos_267___variants___specialization_pow_two","anchor":"Erdos267___erdos_267___variants___specialization_pow_two","docHtml":"\n Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that $\\sum_n \\frac 1 {F_{2^n}}$ is irrational.
\n\n Formal proof provided by AlphaProof\nRef:
\n\n [Go74] Good, I. J.,
\n [BiHo76] Hoggatt, Jr., V. E. and Bicknell, Marjorie,
\n The sum $\\sum_n \\frac 1 {F_{n}}$ itself was proved to be irrational by André-Jeannin.
\n\n Ref: André-Jeannin, Richard,
\n Let $S \\subseteq \\mathbb{R}$ be a set containing no solutions to $a + b = c$.\nMust there be a set $A \\subseteq \\mathbb{R} \\setminus S$ of cardinality continuum such that\n$A + A \\subseteq \\mathbb{R}\\setminus S$?
"},"Erdos949.erdos_949.variants.sidon":{"url":"/FormalConjectures/ErdosProblems/«949»/#Erdos949___erdos_949___variants___sidon","anchor":"Erdos949___erdos_949___variants___sidon","docHtml":"\n Let $S\\sub \\mathbb{R}$ be a Sidon set. Must there be a set $A\\sub \\mathbb{R}∖S$ of cardinality\ncontinuum such that $A + A \\sub \\mathbb{R}∖S$?
"},"Erdos965.erdos_965":{"url":"/FormalConjectures/ErdosProblems/«965»/#Erdos965___erdos_965","anchor":"Erdos965___erdos_965","docHtml":"\n Erdős asks in [Er75b] if for every 2-coloring of ℝ, there is an uncountable set $A ⊆ ℝ$ such that\nall sums $a + b$ for $a, b ∈ A, a ≠ b$ have the same colour.
\n\n In [Ko16] Péter Komjáth constructed a counterexample.\nThe same result was proven independently in [SWCol] by Sokoup and Weiss.
"},"Erdos965.erdos_965.variants.generalization":{"url":"/FormalConjectures/ErdosProblems/«965»/#Erdos965___erdos_965___variants___generalization","anchor":"Erdos965___erdos_965___variants___generalization","docHtml":"\n In fact, in both [Ko16] and [SWCol] a generalized example for $k$-sums is constructed.
"},"Erdos996.fourierPartial":{"url":"/FormalConjectures/ErdosProblems/«996»/#Erdos996___fourierPartial","anchor":"Erdos996___fourierPartial","docHtml":"\n
\n [Er49d] Erdös, P. \"On the strong law of large numbers.\" Transactions of the American Mathematical\nSociety 67.1 (1949): 51-56.
\n\n [Ma66] Matsuyama, Noboru. \"On the strong law of large numbers.\" Tohoku Mathematical Journal,\nSecond Series 18.3 (1966): 259-269.
\n\n Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary\nsequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x,\nlim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?
\n The following theorem is proved in [Ma66].
"},"Erdos688.Erdos688Prop":{"url":"/FormalConjectures/ErdosProblems/«688»/#Erdos688___Erdos688Prop","anchor":"Erdos688___Erdos688Prop","docHtml":"\n Define $\\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$\nfor all primes $n^{\\epsilon_n} < p \\leq n$ such that every integer in $[1,n]$ satisfies at least\none of the congruences $\\equiv a_p \\pmod p$.
"},"Erdos688.epsilonFunction":{"url":"/FormalConjectures/ErdosProblems/«688»/#Erdos688___epsilonFunction","anchor":"Erdos688___epsilonFunction"},"Erdos688.erdos_688.parts.i.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«688»/#Erdos688___erdos_688___parts___i___lower_bound","anchor":"Erdos688___erdos_688___parts___i___lower_bound","docHtml":"\n Estimate $\\epsilon_n$ - lower bound.
"},"Erdos688.erdos_688.parts.i.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«688»/#Erdos688___erdos_688___parts___i___upper_bound","anchor":"Erdos688___erdos_688___parts___i___upper_bound","docHtml":"\n Estimate $\\epsilon_n$ - upper bound.
"},"Erdos688.erdos_688.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«688»/#Erdos688___erdos_688___parts___ii","anchor":"Erdos688___erdos_688___parts___ii","docHtml":"\n In particular, is it true that $\\epsilon_n = o(1)$?
"},"Erdos688.erdos_688.variants.lglglg_over_lglg_is_big_o":{"url":"/FormalConjectures/ErdosProblems/«688»/#Erdos688___erdos_688___variants___lglglg_over_lglg_is_big_o","anchor":"Erdos688___erdos_688___variants___lglglg_over_lglg_is_big_o","docHtml":"\n Erdős claims in [Er80] (p. 106) that it is not difficult to prove\n$\\epsilon_n \\gg \\frac{\\log\\log\\log n}{\\log\\log n}$.
"},"Erdos850.erdos_850":{"url":"/FormalConjectures/ErdosProblems/«850»/#Erdos850___erdos_850","anchor":"Erdos850___erdos_850","docHtml":"\n Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors,\n$x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?
"},"Erdos155.F":{"url":"/FormalConjectures/ErdosProblems/«155»/#Erdos155___F","anchor":"Erdos155___F","docHtml":"\n Let $F(N)$ be the size of the largest Sidon subset of ${1, \\dots, N}$.
"},"Erdos155.erdos_155":{"url":"/FormalConjectures/ErdosProblems/«155»/#Erdos155___erdos_155","anchor":"Erdos155___erdos_155","docHtml":"\n Is it true that for every $k \\geq 1$ we have\n$$\nF(N + k) \\leq F(N) + 1\n$$\nfor all sufficiently large $N$?
"},"Erdos918.erdos_918.parts.i":{"url":"/FormalConjectures/ErdosProblems/«918»/#Erdos918___erdos_918___parts___i","anchor":"Erdos918___erdos_918___parts___i","docHtml":"\n Is there a graph with $\\aleph_2$ vertices and chromatic number $\\aleph_2$ such that every\nsubgraph on $\\aleph_1$ vertices has chromatic number $\\leq\\aleph_0$?
"},"Erdos918.erdos_918.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«918»/#Erdos918___erdos_918___parts___ii","anchor":"Erdos918___erdos_918___parts___ii","docHtml":"\n Is there a graph with $\\aleph_{\\omega+1}$ vertices and chromatic number $\\aleph_1$ such that\nevery subgraph on $\\aleph_\\omega$ vertices has chromatic number $\\leq\\aleph_0$?
"},"Erdos918.erdos_918.variants.all_subgraphs.parts.i":{"url":"/FormalConjectures/ErdosProblems/«918»/#Erdos918___erdos_918___variants___all_subgraphs___parts___i","anchor":"Erdos918___erdos_918___variants___all_subgraphs___parts___i","docHtml":"\n Is there a graph with $\\aleph_2$ vertices and chromatic number $\\aleph_2$ such that every\nsubgraph on $\\aleph_1$ vertices has chromatic number $\\leq\\aleph_0$?
"},"Erdos918.erdos_918.variants.all_subgraphs.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«918»/#Erdos918___erdos_918___variants___all_subgraphs___parts___ii","anchor":"Erdos918___erdos_918___variants___all_subgraphs___parts___ii","docHtml":"\n Is there a graph with $\\aleph_{\\omega+1}$ vertices and chromatic number $\\aleph_1$ such that\nevery subgraph on $\\aleph_\\omega$ vertices has chromatic number $\\leq\\aleph_0$?
"},"Erdos918.erdos_918.variants.erdos_hajnal":{"url":"/FormalConjectures/ErdosProblems/«918»/#Erdos918___erdos_918___variants___erdos_hajnal","anchor":"Erdos918___erdos_918___variants___erdos_hajnal","docHtml":"\n A question of Erd\\H{o}s and Hajnal [ErHa68b], who proved that for every finite $k$\nthere is a graph with chromatic number $\\aleph_1$ and $\\aleph_k$ vertices where each subgraph on\nless than $\\aleph_k$ vertices has chromatic number $\\leq \\aleph_0$.
"},"Erdos918.erdos_918.variants.eq_aleph_0.parts.i":{"url":"/FormalConjectures/ErdosProblems/«918»/#Erdos918___erdos_918___variants___eq_aleph_0___parts___i","anchor":"Erdos918___erdos_918___variants___eq_aleph_0___parts___i","docHtml":"\n In [ErHa69] the questions are stated with $= \\aleph_0$ rather than $\\leq\\aleph_0$. This is\na likely typo since it can be shown that no such graph exists in this case.
\n\n This is the first question with induced subgraphs.
"},"Erdos918.erdos_918.variants.eq_aleph_0_all_subgraphs.parts.i":{"url":"/FormalConjectures/ErdosProblems/«918»/#Erdos918___erdos_918___variants___eq_aleph_0_all_subgraphs___parts___i","anchor":"Erdos918___erdos_918___variants___eq_aleph_0_all_subgraphs___parts___i","docHtml":"\n In [ErHa69] the questions are stated with $= \\aleph_0$ rather than $\\leq\\aleph_0$. This is\na likely typo since it can be shown that no such graph exists in this case.
\n\n This is the first question with all subgraphs.
"},"Erdos918.erdos_918.variants.eq_aleph_0.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«918»/#Erdos918___erdos_918___variants___eq_aleph_0___parts___ii","anchor":"Erdos918___erdos_918___variants___eq_aleph_0___parts___ii","docHtml":"\n In [ErHa69] the questions are stated with $= \\aleph_0$ rather than $\\leq\\aleph_0$. This is\na likely typo since it can be shown that no such graph exists in this case.
\n\n This is the second question with induced subgraphs.
"},"Erdos918.erdos_918.variants.eq_aleph_0_all_subgraphs.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«918»/#Erdos918___erdos_918___variants___eq_aleph_0_all_subgraphs___parts___ii","anchor":"Erdos918___erdos_918___variants___eq_aleph_0_all_subgraphs___parts___ii","docHtml":"\n In [ErHa69] the questions are stated with $= \\aleph_0$ rather than $\\leq\\aleph_0$. This is\na likely typo since it can be shown that no such graph exists in this case.
\n\n This is the second question with all subgraphs.
"},"Erdos203.erdos_203":{"url":"/FormalConjectures/ErdosProblems/«203»/#Erdos203___erdos_203","anchor":"Erdos203___erdos_203","docHtml":"\n Is there an integer $m$ with $(m, 6) = 1$ such that none of $2^k \\cdot 3^\\ell \\cdot m + 1$ are prime,\nfor any $k, \\ell \\ge 0$?
"},"Erdos1060.erdos_1060.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1060»/#Erdos1060___erdos_1060___parts___i","anchor":"Erdos1060___erdos_1060___parts___i","docHtml":"\n The conjecture is about the function $f(n)$ which counts the number of solutions to\n$k\\sigma(k)=n$, where $\\sigma(k)$ is the sum of divisors of $k$. The first bound is that $f(n)$ grows slower\nthan any power of $n^(\\frac{1}{\\log\\log n})$. The second bound is that $f(n)$ is at most a power of\n$\\log n$.
"},"Erdos1060.erdos_1060.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«1060»/#Erdos1060___erdos_1060___parts___ii","anchor":"Erdos1060___erdos_1060___parts___ii","docHtml":"\n Part (ii) of Erdős Problem 1060: bound on the number of $k \\le n$ with $k \\sigma_1(k) = n$.
"},"Erdos975.Erdos975Sum":{"url":"/FormalConjectures/ErdosProblems/«975»/#Erdos975___Erdos975Sum","anchor":"Erdos975___Erdos975Sum","docHtml":"\n Sum of $\\tau(f(n))$ from 0 to ⌊x⌋ for a polynomial $f \\in \\mathbb{Z}[X]$.
\n Here $\\tau$ is the divisor counting function, which is σ 0 in mathlib.\nAlso, for simplicity, we use Nat.floor to convert rational values to natural numbers, instead of\ndealing with negative values.
\n For an irreducible polynomial $f \\in \\mathbb{Z}[x]$ with $f(n) \\ge 1$ for sufficiently large $n$,\ndoes there exists a constant $c = c(f) > 0$ such that\n$\\sum_{n \\le x} \\tau(f(n)) \\approx c \\cdot x \\log x$?
\n\n Note that it is unclear whether the polynomial should have integer coefficients or merely be\ninteger-valued. We assume the former.
"},"Erdos975.erdos_975.variants.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«975»/#Erdos975___erdos_975___variants___upper_bound","anchor":"Erdos975___erdos_975___variants___upper_bound","docHtml":"\n The correctness of the growth rate is shown in [Va39] (lower bound) and [Er52b] (upper bound).
"},"Erdos975.erdos_975.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«975»/#Erdos975___erdos_975___variants___lower_bound","anchor":"Erdos975___erdos_975___variants___lower_bound","docHtml":"\n Lower bound for the growth rate of Erdos975Sum, shown in [Va39].
\n When $f$ is an irreducible quadratic polynomial, the question is answered first by Hooley [Ho63].\nMore compact expression of the constant in terms of Hurwitz class numbers (when $a = 1$)\nis given by McKey in [Mc95], [Mc97], [Mc99].
\n\n TODO: formalize Hurwitz class numbers and the expression of the constant in terms of them.
"},"Erdos975.erdos_975.variants.n2_plus_1_strong":{"url":"/FormalConjectures/ErdosProblems/«975»/#Erdos975___erdos_975___variants___n2_plus_1_strong","anchor":"Erdos975___erdos_975___variants___n2_plus_1_strong","docHtml":"\n More concrete example for $f(n) = n^2 + 1$, where the asymptote is\n$\\sum_{n \\le x} \\tau(n^2 + 1) \\sim \\frac{3}{\\pi} x \\log x + O(x)$. See Tao's blog [T].
"},"Erdos975.erdos_975.variants.n2_plus_1":{"url":"/FormalConjectures/ErdosProblems/«975»/#Erdos975___erdos_975___variants___n2_plus_1","anchor":"Erdos975___erdos_975___variants___n2_plus_1","docHtml":"\n Asymptotics for Erdos975Sum with $f(X) = X^2 + 1$.
\n Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we\n$2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some\nmonochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.
\n\n Prove that, for $r \\ge 3$,\n$$ \\log_{r-1} R_r(n) \\asymp_r n, $$\nwhere $\\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm.
"},"Erdos119.p":{"url":"/FormalConjectures/ErdosProblems/«119»/#Erdos119___p","anchor":"Erdos119___p","docHtml":"\n Let $z_i$ be an infinite sequence of complex numbers such that $|z_i| = 1$ for all $i \\geq 1$.\nFor $n \\geq 1$ let $p_n(z) = \\prod_{i \\leq n} (z - z_i)$.
"},"Erdos119.M":{"url":"/FormalConjectures/ErdosProblems/«119»/#Erdos119___M","anchor":"Erdos119___M","docHtml":"\n Let $M_n = \\max_{|z| = 1} |p_n(z)|$.
"},"Erdos119.erdos_119.parts.i":{"url":"/FormalConjectures/ErdosProblems/«119»/#Erdos119___erdos_119___parts___i","anchor":"Erdos119___erdos_119___parts___i","docHtml":"\n Question 1:
\n\n Is it true that $\\limsup M_n = \\infty$?
\n\n Wagner [Wa80] proved that there is some $c > 0$ with $M_n > (\\log n)^c$ infintely often.
\n\n [Wa80] Wagner, Gerold, On a problem of {E}rdős in {D}iophantine approximation. Bull. London Math. Soc. (1980), 81--88.
"},"Erdos119.erdos_119.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«119»/#Erdos119___erdos_119___parts___ii","anchor":"Erdos119___erdos_119___parts___ii","docHtml":"\n Question 2:
\n\n Is it true that there exists $c > 0$ such that for infinitely many $n$ we have $M_n > n^c$?
\n\n Beck [Be91] proved that there exists some $c > 0$ such that $\\max_{n \\leq N} M_n > N^c$.
\n\n [Be91] Beck, J., The modulus of polynomials with zeros on the unit circle: A problem of Erdős. Annals of Math. (1991), 609-651.
"},"Erdos119.erdos_119.parts.iii":{"url":"/FormalConjectures/ErdosProblems/«119»/#Erdos119___erdos_119___parts___iii","anchor":"Erdos119___erdos_119___parts___iii","docHtml":"\n Question 3:
\n\n Is it true that there exists $c > 0$ such that, for all large $n$, $\\sum_{k \\leq n} M_k > n^{1 + c}$?
"},"Erdos516.OfFiniteOrder":{"url":"/FormalConjectures/ErdosProblems/«516»/#Erdos516___OfFiniteOrder","anchor":"Erdos516___OfFiniteOrder","docHtml":"\n An entire function f is said to be of finite order if there exist numbers c, a ≥ 0\nsuch that for all z, ‖f z‖ ≤ c * rexp (‖z‖ ^ a).
\n Let f = ∑ aₖzⁿₖ be an entire function of finite order such that nₖ / k → ∞.\nThen limsup (fun r => ratio r f) atTop = 1. This is proved in [Fu63].
\n Let f = ∑ aₖzⁿₖ be an entire function such that nₖ > k (log k) ^ (2 + c).\nThen limsup (fun r => ratio r f) atTop = 1. This is proved in [Ko65].
\n Is it true that for all entire functions f = ∑ aₖzⁿₖ such that ∑' 1 / nₖ < ∞,\nlimsup (fun r => ratio r f) atTop = 1?
\n If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞,\nis it true that f assumes every value infinitely often?
\n If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that ∑ 1 / nₖ < ∞,\nthen f assumes every value infinitely often. This theorem is proved in [Bi28].
\n This set contains all solutions (m, P) to the Erdős problem 313.\nA solution is a pair where m is an integer ≥ 2 and P is a non-empty, finite set of\ndistinct prime numbers, such that the sum of the reciprocals of the primes in P equals 1 - 1/m.
\n Are there infinitely many pairs (m, P) where m ≥ 2 is an integer\nand P is a set of distinct primes such that the following equation holds:\n$\\sum_{p \\in P} \\frac{1}{p} = 1 - \\frac{1}{m}$?
\n An integer n is a primary pseudoperfect number if it is the denominator m in a\nsolution (m, P) to the Erdős 313 problem.
\n It is conjectured that the set of primary pseudoperfect numbers is infinite.
"},"Erdos313.erdos_313.variants.exists_at_least_eight_primary_pseudoperfect":{"url":"/FormalConjectures/ErdosProblems/«313»/#Erdos313___erdos_313___variants___exists_at_least_eight_primary_pseudoperfect","anchor":"Erdos313___erdos_313___variants___exists_at_least_eight_primary_pseudoperfect","docHtml":"\n There are at least 8 primary pseudoperfect numbers. The first eight terms of\nA54377 are exhibited together with their explicit\nprime decompositions.
"},"Finset":{"url":"/FormalConjectures/Paper/StrongSensitivityConjecture/#Finset","anchor":"Finset","docHtml":"\n Check validity of block collection (disjoint and sensitive),\nA collection of blocks cB is valid for f at x if the blocks are\ndisjoint and flipping any block changes f(x).
\n Given a non-negative integer $n$, we say $m$ is a separating cardinality of\nsubset sums if, for any set $A$ of $n$ integers, there is some $B\\subseteq A$ of\nsize $\\geq m$ such that subset sums of $B$ can only ever coincide when the\nsubsets have the same cardinality.
"},"Erdos789.subsetSumThreshold":{"url":"/FormalConjectures/ErdosProblems/«789»/#Erdos789___subsetSumThreshold","anchor":"Erdos789___subsetSumThreshold","docHtml":"\n The subset sum threshold $h(n)$, for each positive $n$, is the maximal separating\ncardinality of subset sums for $n$.
"},"Erdos789.erdos_789":{"url":"/FormalConjectures/ErdosProblems/«789»/#Erdos789___erdos_789","anchor":"Erdos789___erdos_789","docHtml":"\n Let $h(n)$ be maximal such that if $A\\subseteq \\mathbb{Z}$ with $\\lvert A\\rvert=n$\nthen there is $B\\subseteq A$ with $\\lvert B\\rvert \\geq h(n)$ such that if\n$a_1+\\cdots+a_r=b_1+\\cdots+b_s$ with $a_i,b_i\\in B$ then $r=s$.
\n\n Estimate $h(n)$.
"},"Erdos789.erdos_789.variants.sq":{"url":"/FormalConjectures/ErdosProblems/«789»/#Erdos789___erdos_789___variants___sq","anchor":"Erdos789___erdos_789___variants___sq","docHtml":"\n Let $h(n)$ be maximal such that if $A\\subseteq \\mathbb{Z}$ with $\\lvert A\\rvert=n$\nthen there is $B\\subseteq A$ with $\\lvert B\\rvert \\geq h(n)$ such that if\n$a_1+\\cdots+a_r=b_1+\\cdots+b_s$ with $a_i,b_i\\in B$ then $r=s$.
\n\n Is $h(n) = \\Theta(\\sqrt{n})$?
"},"Erdos789.erdos_789.variants.isBigO_sq":{"url":"/FormalConjectures/ErdosProblems/«789»/#Erdos789___erdos_789___variants___isBigO_sq","anchor":"Erdos789___erdos_789___variants___isBigO_sq","docHtml":"\n Straus [Str66] proved that $h(n) \\ll \\sqrt{n}$.
"},"Erdos789.erdos_789.variants.sq_isBigO":{"url":"/FormalConjectures/ErdosProblems/«789»/#Erdos789___erdos_789___variants___sq_isBigO","anchor":"Erdos789___erdos_789___variants___sq_isBigO","docHtml":"\n By the solved variant erdos_789.variants.isBigO_sq, in order to prove\nerdos_789.variants.sq it suffices to show $\\sqrt{n}=O(h(n))$.
\n Let $h(n)$ be maximal such that if $A\\subseteq \\mathbb{Z}$ with $\\lvert A\\rvert=n$\nthen there is $B\\subseteq A$ with $\\lvert B\\rvert \\geq h(n)$ such that if\n$a_1+\\cdots+a_r=b_1+\\cdots+b_s$ with $a_i,b_i\\in B$ then $r=s$.
\n\n Is $h(n) = \\Theta((n\\log(n)))^{1/3})$?
"},"Erdos789.erdos_789.variants.cube_root_linearithmic_isBigO":{"url":"/FormalConjectures/ErdosProblems/«789»/#Erdos789___erdos_789___variants___cube_root_linearithmic_isBigO","anchor":"Erdos789___erdos_789___variants___cube_root_linearithmic_isBigO","docHtml":"\n Erdős [Er62c] and Choi [Ch74b] proved that $(n\\log(n))^{1/3}\\ll h(n)$.
"},"Erdos789.erdos_789.variants.isBigO_cube_root_linearithmic":{"url":"/FormalConjectures/ErdosProblems/«789»/#Erdos789___erdos_789___variants___isBigO_cube_root_linearithmic","anchor":"Erdos789___erdos_789___variants___isBigO_cube_root_linearithmic","docHtml":"\n By the solved variant erdos_789.variants.cube_root_linearithmic_isBigO, in order to prove\nerdos_789.variants.cube_root_linarithmic it suffices to show $h(n) = O((n\\log(n))^{1/3})$.
\n If $n$ distinct points in $\\mathbb{R}^2$ form a convex polygon then some vertex has at least\n$\\lfloor\\frac{n}{2}\\rfloor$ different distances to other vertices.
"},"Erdos454.f":{"url":"/FormalConjectures/ErdosProblems/«454»/#Erdos454___f","anchor":"Erdos454___f","docHtml":"\n Define f n to be the minimum of (n + i).nth Prime + (n - i).nth Prime over 0 < i < n.
\n Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤?
\nlimsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop ≥ 2, and this is proved in [Po79].
\nErdős Problem 42: Let M ≥ 1 and N be sufficiently large in terms of M. Is it true that for every\nmaximal Sidon set A ⊆ {1,…,N} there is another Sidon set B ⊆ {1,…,N} of size M such that\n(A - A) ∩ (B - B) = {0}?
\n This was proved for all $M$ by GPT 5.5 Pro (prompted by Sandhu), see discussion thread for more details.
"},"Erdos42.erdos_42.variants.constructive":{"url":"/FormalConjectures/ErdosProblems/«42»/#Erdos42___erdos_42___variants___constructive","anchor":"Erdos42___erdos_42___variants___constructive","docHtml":"\n A variant asking for explicit bounds on how large N needs to be in terms of M.
\n\n This version provides a constructive function f such that for all M ≥ 1 and N ≥ f(M),\nevery maximal Sidon set A ⊆ {1,…,N} has another Sidon set B ⊆ {1,…,N} of size M with\ndisjoint difference sets (apart from 0).
"},"Erdos42.example_maximal_sidon":{"url":"/FormalConjectures/ErdosProblems/«42»/#Erdos42___example_maximal_sidon","anchor":"Erdos42___example_maximal_sidon","docHtml":"\n The set {1, 2, 4} is a maximal Sidon set in {1, ..., 4}.
\n The difference set of {1, 2, 4} is {0, 1, 2, 3}.
\n For any maximal Sidon set, the difference set contains 0.
"},"Erdos442.Real.maxLogOne":{"url":"/FormalConjectures/ErdosProblems/«442»/#Erdos442___Real___maxLogOne","anchor":"Erdos442___Real___maxLogOne","docHtml":"\n The function $\\operatorname{Log} x := \\max{log x, 1}$.
"},"Erdos442.Set.bddProdUpper":{"url":"/FormalConjectures/ErdosProblems/«442»/#Erdos442___Set___bddProdUpper","anchor":"Erdos442___Set___bddProdUpper","docHtml":"\n If A be a set of natural numbers and let x be real, then\nA.bddProdUpper x is the finite upper-triangular set of pairs\nof elements of A that are ≤ x. Specifically, it is the set\n{(n, m) | n ∈ A, n ≤ x, m ∈ A, m ≤ x, n < m}
\n Let $\\operatorname{Log} x := \\max{\\log x, 1}$,\n$\\operatorname{Log}
\n Tao [Ta24b] has shown this is false.
\n\n [Ta24b] Tao, T.,
\n Note: the informal and formal statements follow the solution paper https://arxiv.org/pdf/2407.04226
"},"Erdos442.erdos_442.variants.tao":{"url":"/FormalConjectures/ErdosProblems/«442»/#Erdos442___erdos_442___variants___tao","anchor":"Erdos442___erdos_442___variants___tao","docHtml":"\n Tao resolved erdos_442 in the negative in Theorem 1 of https://arxiv.org/pdf/2407.04226.\nThe following is a formalisation of that theorem with $C_0 = 1$.
\n\n Let $\\operatorname{Log} x := \\max{\\log x, 1}$,\n$\\operatorname{Log}_2x = \\operatorname{Log} (\\operatorname{Log} x)$, and\n$\\operatorname{Log}
\n Is it true that $\\frac{R(n+1)}{R(n)}\\geq 1+c$ for some constant $c>0$, for all large $n$?
"},"Erdos812.erdos_812.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«812»/#Erdos812___erdos_812___parts___ii","anchor":"Erdos812___erdos_812___parts___ii","docHtml":"\n Is it true that $R(n+1)-R(n) \\gg n^2$?
"},"Erdos812.erdos_812.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«812»/#Erdos812___erdos_812___variants___lower_bound","anchor":"Erdos812___erdos_812___variants___lower_bound","docHtml":"\n Burr, Erdős, Faudree, and Schelp [BEFS89] proved that $R(n+1)-R(n) \\geq 4n-8$ for all $n\\geq 2$.
"},"Erdos377.sumInvPrimesNotDvdCentralBinom":{"url":"/FormalConjectures/ErdosProblems/«377»/#Erdos377___sumInvPrimesNotDvdCentralBinom","anchor":"Erdos377___sumInvPrimesNotDvdCentralBinom","docHtml":"\n The sum of the inverses of all primes smaller than $n$, which don't divide the central\nbinom coefficient.
"},"Erdos377.erdos_377":{"url":"/FormalConjectures/ErdosProblems/«377»/#Erdos377___erdos_377","anchor":"Erdos377___erdos_377","docHtml":"\n Is there some absolute constant $C > 0$ such that\n$$\n\\sum_{p \\leq n} 1_{p\\nmid {2n \\choose n}}\\frac{1}{p} \\leq C\n$$\nfor all $n$?
"},"Erdos377.erdos_377.variants.limit.i":{"url":"/FormalConjectures/ErdosProblems/«377»/#Erdos377___erdos_377___variants___limit___i","anchor":"Erdos377___erdos_377___variants___limit___i","docHtml":"\n Erdos, Graham, Ruzsa, and Straus proved that if\n$$\nf(n) = \\sum_{p \\leq n} 1_{p\\nmid {2n \\choose n}}\\frac{1}{p}\n$$\nand\n$$\n\\gamma_0 = \\sum_{k = 2}^{\\infty} \\frac{\\log k}{2^k}\n$$\nthen\n$$\n\\lim_{x\\to\\infty} \\frac{1}{x}\\sum_{n\\leq x} f(n) = \\gamma_0\n$$
\n\n [EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G.,
\n Erdos, Graham, Ruzsa, and Straus proved that if\n$$\nf(n) = \\sum_{p \\leq n} 1_{p\\nmid {2n \\choose n}}\\frac{1}{p}\n$$\nand\n$$\n\\gamma_0 = \\sum_{k = 2}^{\\infty} \\frac{\\log k}{2^k}\n$$\nthen\n$$\n\\lim_{x\\to\\infty} \\frac{1}{x}\\sum_{n\\leq x} f(n)^2 = \\gamma_0^2\n$$
\n\n [EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G.,
\n Erdos, Graham, Ruzsa, and Straus proved that if\n$$\nf(n) = \\sum_{p \\leq n} 1_{p\\nmid {2n \\choose n}}\\frac{1}{p}\n$$\nand\n$$\n\\gamma_0 = \\sum_{k = 2}^{\\infty} \\frac{\\log k}{2^k}\n$$\nthen for almost all integers $f(m) = \\gamma_0 + o(1)$.
\n\n [EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G.,
\n Erdos, Graham, Ruzsa, and Straus proved that if\n$$\nf(n) = \\sum_{p \\leq n} 1_{p\\nmid {2n \\choose n}}\\frac{1}{p}\n$$\nthen there is some constant $c < 1$ such that for all large $n$\n$$\nf(n) \\leq c \\log\\log n.\n$$
\n\n [EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G.,
\n Let $P(n, k)$ be the largest prime factor of $\\binom{n}{k}$.
"},"Erdos683.erdos_683":{"url":"/FormalConjectures/ErdosProblems/«683»/#Erdos683___erdos_683","anchor":"Erdos683___erdos_683","docHtml":"\n There exists $c > 0$ such that $P(n, k) > \\min{n-k+1, k^{1 + c}}$ for all $0 < k < n$.}
"},"Erdos683.erdos_683.variant.sylvester_schur":{"url":"/FormalConjectures/ErdosProblems/«683»/#Erdos683___erdos_683___variant___sylvester_schur","anchor":"Erdos683___erdos_683___variant___sylvester_schur","docHtml":"\n Sylvester and Schur [Er34] proved that $P(n, k) > k$ for $k \\le n/2$.
"},"Erdos683.erdos_683.variant.erdos_log":{"url":"/FormalConjectures/ErdosProblems/«683»/#Erdos683___erdos_683___variant___erdos_log","anchor":"Erdos683___erdos_683___variant___erdos_log","docHtml":"\n Erdos [Er55d] improved this to $P(n, k) \\gg k \\log k $ for $k \\le n/2$.
"},"Erdos683.erdos_683.variant.exp_sqrt":{"url":"/FormalConjectures/ErdosProblems/«683»/#Erdos683___erdos_683___variant___exp_sqrt","anchor":"Erdos683___erdos_683___variant___exp_sqrt","docHtml":"\n Standard heuristics suggest that $P(n, k) > e^{c\\sqrt{k}}$ for some constant $c > 0$.
"},"Erdos1167.erdos_1167":{"url":"/FormalConjectures/ErdosProblems/«1167»/#Erdos1167___erdos_1167","anchor":"Erdos1167___erdos_1167","docHtml":"\nErdős Problem 1167. Let $r \\geq 2$ be finite and $\\lambda$ be an infinite\ncardinal. Let $\\kappa_\\alpha$ be cardinals for all $\\alpha < \\gamma$. Is it true\nthat\n$$2^\\lambda \\to (\\kappa_\\alpha + 1)
\n A problem of Erdős, Hajnal, and Rado.
"},"Erdos1167.erdos_1167.variants.finite_targets":{"url":"/FormalConjectures/ErdosProblems/«1167»/#Erdos1167___erdos_1167___variants___finite_targets","anchor":"Erdos1167___erdos_1167___variants___finite_targets","docHtml":"\nFinite-target case. When all $\\kappa_\\alpha$ are finite, $\\kappa_\\alpha + 1$\nis the ordinary natural-number successor. Special case of erdos_1167.
\nBinary-color case. The $\\gamma = 2$ specialization (two color classes).
"},"Erdos1167.erdos_1167.variants.infinite_targets":{"url":"/FormalConjectures/ErdosProblems/«1167»/#Erdos1167___erdos_1167___variants___infinite_targets","anchor":"Erdos1167___erdos_1167___variants___infinite_targets","docHtml":"\nInfinite-target case. When all $\\kappa_\\alpha \\geq \\aleph_0$ are infinite,\n$\\kappa_\\alpha + 1 = \\kappa_\\alpha$, so the hypothesis simplifies to a \"pure\"\nstepping-down lemma:\n$$2^\\lambda \\to (\\kappa_\\alpha)
\n$r = 2$ case. The stepping-down from 3-uniform to 2-uniform partition\nrelations: $2^\\lambda \\to (\\kappa_\\alpha + 1)
\n
\nA129515
\n\n Are there infinitely many pairs of integers $n < m$ such that $\\binom{2n}{n}$\nand $\\binom{2m}{m}$ have the same set of prime divisors?
"},"Erdos730.erdos_730.variants.explicit_pairs":{"url":"/FormalConjectures/ErdosProblems/«730»/#Erdos730___erdos_730___variants___explicit_pairs","anchor":"Erdos730___erdos_730___variants___explicit_pairs","docHtml":"\n For example, $(87,88)$ and $(607,608)$ are such pairs.
"},"Erdos730.erdos_730.variants.delta_ne_one":{"url":"/FormalConjectures/ErdosProblems/«730»/#Erdos730___erdos_730___variants___delta_ne_one","anchor":"Erdos730___erdos_730___variants___delta_ne_one","docHtml":"\n There are examples where $(n, m) ∈ S$ with $m ≠ n + 1$.
\n\n (Found by AlphaProof, although it was implicit already in [A129515])
"},"Erdos350.DistinctSubsetSums":{"url":"/FormalConjectures/ErdosProblems/«350»/#Erdos350___DistinctSubsetSums","anchor":"Erdos350___DistinctSubsetSums","docHtml":"\n The predicate that all (finite) subsets of A have distinct sums.
\n The predicate that all (finite) subsets of A have distinct sums, decidable version
\n Small sanity check: the two predicates are saying the same thing.
"},"Erdos350.erdos_350":{"url":"/FormalConjectures/ErdosProblems/«350»/#Erdos350___erdos_350","anchor":"Erdos350___erdos_350","docHtml":"\n If A ⊂ ℕ is a finite set of integers all of whose subset sums are distinct then ∑ n ∈ A, 1/n < 2.\nProved by Ryavec.
\n This was proved by Ryavec, who did not appear to ever publish the proof. Ryavec's proof is\nreproduced in [BeEr74]. More generally, Ryavec's proof delivers that\n$\\sum_{n\\in A}\\frac{1}{n}\\leq 2-2^{1-\\lvert A\\rvert},$ with equality if and only if\n$A={1,2,\\ldots,2^k}$.
\n\n This was formalized in Lean by Alexeev using Aristotle.
"},"Erdos350.erdos_350.variants.strengthening":{"url":"/FormalConjectures/ErdosProblems/«350»/#Erdos350___erdos_350___variants___strengthening","anchor":"Erdos350___erdos_350___variants___strengthening","docHtml":"\n If A ⊂ ℕ is a finite set of integers all of whose subset sums are distinct then ∑ n ∈ A, 1/n^s < 1/(1 - 2^(-s)), for any s > 0.\nProved by Hanson, Steele, and Stenger [HSS77].
\n We exlude here the case s = 0, because in the informal formulation then the right hand side is to be interpreted as ∞, while the left hand side counts the elements in A.
\n Are there any $2$-full $n$ such that $n+1$ is $3$-full?
"},"Erdos366.exists_three_full_then_two_full":{"url":"/FormalConjectures/ErdosProblems/«366»/#Erdos366___exists_three_full_then_two_full","anchor":"Erdos366___exists_three_full_then_two_full","docHtml":"\n Note that $8$ is $3$-full and $9$ is 2-full.
"},"Erdos366.erdos_366.variants.three_two":{"url":"/FormalConjectures/ErdosProblems/«366»/#Erdos366___erdos_366___variants___three_two","anchor":"Erdos366___erdos_366___variants___three_two","docHtml":"\n Are there infinitely many 3-full $n$ such that $n+1$ is 2-full?
"},"Erdos366.erdos_366.variants.weaker":{"url":"/FormalConjectures/ErdosProblems/«366»/#Erdos366___erdos_366___variants___weaker","anchor":"Erdos366___erdos_366___variants___weaker","docHtml":"\n Are there any consecutive pairs of $3$-full integers?
"},"Erdos120.Erdos120For":{"url":"/FormalConjectures/ErdosProblems/«120»/#Erdos120___Erdos120For","anchor":"Erdos120___Erdos120For","docHtml":"\n There exists a set $E \\subseteq \\mathbb{R}$, dependent on set $A \\subseteq \\mathbb{R}$,\nof positive measure which does not contain any set of the shape $a * A + b$\nfor some $a,b \\in \\mathbb{R}$ and $a \\neq 0$?
"},"Erdos120.erdos_120":{"url":"/FormalConjectures/ErdosProblems/«120»/#Erdos120___erdos_120","anchor":"Erdos120___erdos_120","docHtml":"\n Let $A \\subseteq \\mathbb{R}$ be an infinite set. Must there be a set $E \\subseteq \\mathbb{R}$\nof positive measure which does not contain any set of the shape $a * A + b$\nfor some $a,b \\in \\mathbb{R}$ and $a \\neq 0$?
"},"Erdos120.erdos_120.variants.finite_set":{"url":"/FormalConjectures/ErdosProblems/«120»/#Erdos120___erdos_120___variants___finite_set","anchor":"Erdos120___erdos_120___variants___finite_set","docHtml":"\n Steinhaus [St20] has proved Erdős 120 to be false whenever $A$ is a finite set.
"},"Erdos74.SimpleGraph.edgeDistancesToBipartite":{"url":"/FormalConjectures/ErdosProblems/«74»/#Erdos74___SimpleGraph___edgeDistancesToBipartite","anchor":"Erdos74___SimpleGraph___edgeDistancesToBipartite","docHtml":"\n For a given subgraph A, this is the set of all numbers k such that A can be made\nbipartite by deleting k edges.
\n The set of edge distances to a bipartite graph is always non-empty because deleting all edges\nfrom a graph makes it bipartite.
"},"Erdos74.SimpleGraph.minEdgeDistToBipartite":{"url":"/FormalConjectures/ErdosProblems/«74»/#Erdos74___SimpleGraph___minEdgeDistToBipartite","anchor":"Erdos74___SimpleGraph___minEdgeDistToBipartite","docHtml":"\n The minimum number of edges that must be deleted from a subgraph A to make it bipartite.
\n For a graph G and a number n, this is the set of minEdgeDistToBipartite A for all\ninduced subgraphs A of G on n vertices.
\n The set of minimum edge distances to bipartite for subgraphs of size n is bounded above.\nA graph on n vertices has at most n choose 2 edges, and deleting all of them\nmakes the graph bipartite, providing a straightforward upper bound.
\n For a given graph $G$ and size $n$, this defines the smallest number $k$\nsuch that any subgraph of $G$ on $n$ vertices can be made bipartite by deleting\nat most $k$ edges.
\n\n This value is optimal because it is the maximum of minEdgeDistToBipartite taken\nover all $n$-vertex subgraphs. This means there exists at least one $n$-vertex\nsubgraph that requires exactly this many edge deletions.\nThis is Definition 3.1 in [EHS82].
\n [EHS82] Erdős, P. and Hajnal, A. and Szemerédi, E.,\n
\n Let $f(n)\\to \\infty$ possibly very slowly.\nIs there a graph of infinite chromatic number such that every finite subgraph on $n$\nvertices can be made bipartite by deleting at most $f(n)$ edges?
"},"Erdos74.erdos_74.variants.sqrt":{"url":"/FormalConjectures/ErdosProblems/«74»/#Erdos74___erdos_74___variants___sqrt","anchor":"Erdos74___erdos_74___variants___sqrt","docHtml":"\n Is there a graph of infinite chromatic number such that every finite subgraph on $n$\nvertices can be made bipartite by deleting at most $\\sqrt{n}$ edges?
"},"Erdos75.erdos_75":{"url":"/FormalConjectures/ErdosProblems/«75»/#Erdos75___erdos_75","anchor":"Erdos75___erdos_75","docHtml":"\n Is there a graph of chromatic number ℵ_ 1 with ℵ_ 1 vertices such that for all\nε > 0, if n is sufficiently large and H is a subgraph on n vertices,\nthen H contains an independent set of size > n ^ (1 - ε)?
\n The predicate that graph $G$ with chromatic number $k$ is such that every vertex is critical, yet\nevery critical set of edges has size $>r$
"},"Erdos944.erdos_944":{"url":"/FormalConjectures/ErdosProblems/«944»/#Erdos944___erdos_944","anchor":"Erdos944___erdos_944","docHtml":"\n Let $k \\ge 4$ and $r\\ge 1$. Must there exist a graph $G$ with chromatic number $k$\nsuch that every vertex is critical, yet every critical set of edges has size $>r$?
"},"Erdos944.erdos_944.variants.dirac_conjecture":{"url":"/FormalConjectures/ErdosProblems/«944»/#Erdos944___erdos_944___variants___dirac_conjecture","anchor":"Erdos944___erdos_944___variants___dirac_conjecture","docHtml":"\n Let $k \\ge 4$. Must there exist a graph $G$ with chromatic number $k$\nsuch that every vertex is critical, yet every critical set of edges has size $>1$?
\n\n This was conjectured by Dirac in 1970.
"},"Erdos944.erdos_944.variants.dirac_conjecture.k_eq_5":{"url":"/FormalConjectures/ErdosProblems/«944»/#Erdos944___erdos_944___variants___dirac_conjecture___k_eq_5","anchor":"Erdos944___erdos_944___variants___dirac_conjecture___k_eq_5","docHtml":"\n Dirac's conjecture was proved, for $k=5$: There exists a graph $G$ with chromatic number $5$, such\nthat every vertex is critical, yet every critical set of edges has size $>1$, or in other words:\nhas no critical edge.
\n\n [Br92] Brown, Jason I., A vertex critical graph without critical edges. Discrete Math. (1992), 99--101
"},"Erdos944.erdos_944.variants.dirac_conjecture.k_sub_one_not_prime":{"url":"/FormalConjectures/ErdosProblems/«944»/#Erdos944___erdos_944___variants___dirac_conjecture___k_sub_one_not_prime","anchor":"Erdos944___erdos_944___variants___dirac_conjecture___k_sub_one_not_prime","docHtml":"\n Lattanzio [La02] proved there exist $k$-critical graphs without critical edges for all $k$ such that\n$k - 1$ is not prime.
\n\n [La02] Lattanzio, John J., A note on a conjecture of {D}irac. Discrete Math. (2002), 323--330
"},"Erdos944.erdos_944.variants.dirac_conjecture.k_ge_five":{"url":"/FormalConjectures/ErdosProblems/«944»/#Erdos944___erdos_944___variants___dirac_conjecture___k_ge_five","anchor":"Erdos944___erdos_944___variants___dirac_conjecture___k_ge_five","docHtml":"\n Jensen [Je02] gave an construction for $k$-critical graphs without any critical edges for all $k ≥ 5$.
\n\n [Je02] Jensen, Tommy R., Dense critical and vertex-critical graphs. Discrete Math. (2002), 63--84.
"},"Erdos944.erdos_944.variants.dirac_conjecture.k_eq_four":{"url":"/FormalConjectures/ErdosProblems/«944»/#Erdos944___erdos_944___variants___dirac_conjecture___k_eq_four","anchor":"Erdos944___erdos_944___variants___dirac_conjecture___k_eq_four","docHtml":"\n The case $k=4$ and $r=1$ remains open: Are there $4$-critical graphs without any critical edges?
"},"Erdos944.erdos_944.variants.large_k_for_any_r":{"url":"/FormalConjectures/ErdosProblems/«944»/#Erdos944___erdos_944___variants___large_k_for_any_r","anchor":"Erdos944___erdos_944___variants___large_k_for_any_r","docHtml":"\n Martinsson and Steiner [MaSt25] proved for every $r \\ge 1$ if $k$ is sufficiently large, depending\non $r$, there exist a graph $G$ with chromatic number $k$ such that every vertex is critical,\nyet every critical set of edges has size $>r$.
\n\n [MaSt25] Martinsson, Anders and Steiner, Raphael, Vertex-critical graphs far from edge-criticality. Combin. Probab. Comput. (2025), 151--157
"},"Erdos972.primeSet":{"url":"/FormalConjectures/ErdosProblems/«972»/#Erdos972___primeSet","anchor":"Erdos972___primeSet","docHtml":"\n The set of primes p such that Nat.floor (α * p) is also prime.
\nErdős problem 972.\nLet $\\alpha > 1$ be irrational. Are there infinitely many primes $p$\nsuch that $\\lfloor p\\alpha \\rfloor$ is also prime?
"},"Erdos887.erdos_887.parts.i":{"url":"/FormalConjectures/ErdosProblems/«887»/#Erdos887___erdos_887___parts___i","anchor":"Erdos887___erdos_887___parts___i","docHtml":"\n Is there an absolute constant $K$ such that, for every $C > 0$, if $n$ is sufficiently large then\n$n$ has at most $K$ divisors in $(n^{\\frac{1}{2}}, n^{\\frac{1}{2}} + C n^{\\frac{1}{4}})$.
"},"Erdos887.erdos_887.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«887»/#Erdos887___erdos_887___parts___ii","anchor":"Erdos887___erdos_887___parts___ii","docHtml":"\n Is there an absolute constant $K$ such that, for every $C > 0$, if $n$ is sufficiently large then\n$n$ has at most $K$ divisors in $(n^{\\frac{1}{2}}, n^{\\frac{1}{2}} + C n^{\\frac{1}{4}})$.
"},"Erdos887.erdos_887.variants.rosenfeld_infinite":{"url":"/FormalConjectures/ErdosProblems/«887»/#Erdos887___erdos_887___variants___rosenfeld_infinite","anchor":"Erdos887___erdos_887___variants___rosenfeld_infinite","docHtml":"\n A question of Erdős and Rosenfeld, who proved that there are infinitely many $n$ with (at least)\n$4$ divisors in $(n^{\\frac{1}{2}}, n^{\\frac{1}{2}} + cn^{\\frac{1}{4}})$.
"},"Erdos887.erdos_887.variants.rosenfeld_4":{"url":"/FormalConjectures/ErdosProblems/«887»/#Erdos887___erdos_887___variants___rosenfeld_4","anchor":"Erdos887___erdos_887___variants___rosenfeld_4","docHtml":"\n Erdős and Rosenfeld, ask whether $4$ is the best possible $K$ for the infinitude of $n$\nwith (at least) $K$ divisors in $(n^{\\frac{1}{2}}, n^{\\frac{1}{2}} + n^{\\frac{1}{4}})$.
"},"Erdos653.erdos_653":{"url":"/FormalConjectures/ErdosProblems/«653»/#Erdos653___erdos_653","anchor":"Erdos653___erdos_653","docHtml":"\n Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ and let $R(x_i)=#{ \\lvert x_j-x_i\\rvert : j\\neq i}$,\nwhere the points are ordered such that\n$$R(x_1)\\leq \\cdots \\leq R(x_n).$$\nLet $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that\n$g(n) \\geq (1-o(1))n$?
"},"Erdos1065.erdos_1065.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1065»/#Erdos1065___erdos_1065___parts___i","anchor":"Erdos1065___erdos_1065___parts___i","docHtml":"\n Are there infinitely many primes $p$ such that $p = 2^k * q + 1$\nfor some prime $q$ and $k ≥ 0$?
\n\n This is mentioned as B46\nin Unsolved Problems in Number Theory\nby
\n Are there infinitely many primes $p$ such that $p = 2^k 3^l q + 1$\nfor some prime $q$ and $k ≥ 0$, $l ≥ 0$?
"},"Erdos389.erdos_389":{"url":"/FormalConjectures/ErdosProblems/«389»/#Erdos389___erdos_389","anchor":"Erdos389___erdos_389","docHtml":"\n Is it true that for every $n \\geq 1$ there is a $k$ such that\n$$\nn(n + 1) \\cdots (n + k - 1) \\mid (n + k) \\cdots (n + 2k - 1)?\n$$
"},"Erdos389.erdos_389.variants.mehta_four":{"url":"/FormalConjectures/ErdosProblems/«389»/#Erdos389___erdos_389___variants___mehta_four","anchor":"Erdos389___erdos_389___variants___mehta_four","docHtml":"\n Bhavik Mehta has computed the minimal such $k$ for $1 \\leq n \\leq 18$.\nFor example, the minimal $k$ for $n = 4$ is $207$.
"},"Erdos50.IsDistributionOfPhiRatio":{"url":"/FormalConjectures/ErdosProblems/«50»/#Erdos50___IsDistributionOfPhiRatio","anchor":"Erdos50___IsDistributionOfPhiRatio","docHtml":"\n A function $f : \\mathbb{R} \\to \\mathbb{R}$ is the asymptotic distribution function of the values\nof $\\varphi(n)/n$ if for all $c \\in [0, 1]$, the natural density of ${n : \\varphi(n) < cn}$\nexists and equals $f(c)$.
"},"Erdos50.IsPurelySingular":{"url":"/FormalConjectures/ErdosProblems/«50»/#Erdos50___IsPurelySingular","anchor":"Erdos50___IsPurelySingular","docHtml":"\n A monotone function $f : \\mathbb{R} \\to \\mathbb{R}$ is purely singular (or singular continuous)\nif it is continuous and its derivative equals zero almost everywhere with respect to Lebesgue\nmeasure.
"},"Erdos50.erdos_50_schoenberg":{"url":"/FormalConjectures/ErdosProblems/«50»/#Erdos50___erdos_50_schoenberg","anchor":"Erdos50___erdos_50_schoenberg","docHtml":"\n Schoenberg [Sch38] proved that the asymptotic distribution function of $\\varphi(n)/n$ exists.\nThat is, for any $c \\in [0, 1]$, the proportion of integers $n \\le N$ satisfying $\\varphi(n)/n < c$\napproaches a limit as $N \\to \\infty$. This limit function is the cumulative distribution function\nof the values of $\\varphi(n)/n$.
"},"Erdos50.erdos_50_singular":{"url":"/FormalConjectures/ErdosProblems/«50»/#Erdos50___erdos_50_singular","anchor":"Erdos50___erdos_50_singular","docHtml":"\n Erdős [Er95] proved that the distribution function of $\\varphi(n)/n$ is purely singular: it is\ncontinuous, but its derivative is zero almost everywhere.
"},"Erdos50.erdos_50":{"url":"/FormalConjectures/ErdosProblems/«50»/#Erdos50___erdos_50","anchor":"Erdos50___erdos_50","docHtml":"\n Let $f$ be the asymptotic distribution function of $\\varphi(n)/n$, so that for each $c \\in [0,1]$,\n$f(c)$ is the natural density of ${n : \\varphi(n) < cn}$. Is it true that there is no $x$ such\nthat the derivative $f'(x)$ exists and is positive?
"},"Erdos417.erdos_417.parts.i":{"url":"/FormalConjectures/ErdosProblems/«417»/#Erdos417___erdos_417___parts___i","anchor":"Erdos417___erdos_417___parts___i","docHtml":"\n Let$$V'(x)=#{\\phi(m) : 1\\leq m\\leq x}$$and$$V(x)=#{\\phi(m) \\leq x : 1\\leq m}.$$\nDoes $\\lim V(x)/V'(x)$ exist?
\n\n Formalization note: We formalize the limit of the inverse fraction V'(x)/V(x)\nto ensure the limit is finite (bounded between 0 and 1).
"},"Erdos417.erdos_417.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«417»/#Erdos417___erdos_417___parts___ii","anchor":"Erdos417___erdos_417___parts___ii","docHtml":"\n Is it $>1$?
"},"Erdos355.erdos_355":{"url":"/FormalConjectures/ErdosProblems/«355»/#Erdos355___erdos_355","anchor":"Erdos355___erdos_355","docHtml":"\n Is there a lacunary sequence $A\\subseteq \\mathbb{N}$ (so that $A={a_1 < \\cdots}$ and\nthere exists some $\\lambda > 1$ such that $a_{n+1}/a_n\\geq \\lambda$ for all $n\\geq 1$) such that\n$$\\left{ \\sum_{a\\in A'}\\frac{1}{a} : A'\\subseteq A\\textrm{ finite}\\right}$$\ncontain all rationals in some open interval?
\n\n Bleicher and Erdős conjectured the answer is no.
\n\n In fact the answer is yes, with any lacunarity constant $\\lambda\\in (1,2)$ (though not $\\lambda=2$),\nas proved by van Doorn and Kova\\v{c} [DoKo25].
\n\n This was formalized in Lean by van Doorn using Aristotle.
"},"Erdos1055.IsOfClass":{"url":"/FormalConjectures/ErdosProblems/«1055»/#Erdos1055___IsOfClass","anchor":"Erdos1055___IsOfClass","docHtml":"\n A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are\n$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor\nof $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.
"},"Erdos1055.exists_p":{"url":"/FormalConjectures/ErdosProblems/«1055»/#Erdos1055___exists_p","anchor":"Erdos1055___exists_p","docHtml":"\n A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are\n$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor\nof $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nShow that for each $r$ there exists a prime $p$ of class $r$.
"},"Erdos1055.p":{"url":"/FormalConjectures/ErdosProblems/«1055»/#Erdos1055___p","anchor":"Erdos1055___p","docHtml":"\n A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are\n$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor\nof $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nLet $p_r$ is the least prime in class $r$.
"},"Erdos1055.erdos_1055":{"url":"/FormalConjectures/ErdosProblems/«1055»/#Erdos1055___erdos_1055","anchor":"Erdos1055___erdos_1055","docHtml":"\n A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are\n$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor\nof $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nAre there infinitely many primes in each class?
"},"Erdos1055.erdos_1055.variants.erdos_limit":{"url":"/FormalConjectures/ErdosProblems/«1055»/#Erdos1055___erdos_1055___variants___erdos_limit","anchor":"Erdos1055___erdos_1055___variants___erdos_limit","docHtml":"\n A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are\n$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor\nof $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nIf $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?\nErdos conjectured that this tends to infinity.
"},"Erdos1055.erdos_1055.variants.selfridge_limit":{"url":"/FormalConjectures/ErdosProblems/«1055»/#Erdos1055___erdos_1055___variants___selfridge_limit","anchor":"Erdos1055___erdos_1055___variants___selfridge_limit","docHtml":"\n A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are\n$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor\nof $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nIf $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?\nSelfridge conjectured that this is bounded.
"},"Erdos1141.Erdos1141Prop":{"url":"/FormalConjectures/ErdosProblems/«1141»/#Erdos1141___Erdos1141Prop","anchor":"Erdos1141___Erdos1141Prop","docHtml":"\n The property that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2 < n$.
"},"Erdos1141.erdos_1141":{"url":"/FormalConjectures/ErdosProblems/«1141»/#Erdos1141___erdos_1141","anchor":"Erdos1141___erdos_1141","docHtml":"\n Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2 < n$?
\n\n In [Va99] it is asked whether $968$ is the largest integer with this property, but this is an\nerror, since for example $968-9=7\\cdot 137$.
\n\n The list of $n$ satisfying the given property is [A214583] in the OEIS. The largest known such $n$\nis $1722$.
\n\n The answer is negative: [APSSV26b] proves a stronger finiteness theorem, deducing it from\nPollack [Po17]. Oriike [Or26] formalised the deduction in Lean.
"},"Erdos375.Erdos375Prop":{"url":"/FormalConjectures/ErdosProblems/«375»/#Erdos375___Erdos375Prop","anchor":"Erdos375___Erdos375Prop","docHtml":"\n This is a proposition saying that for any n ≥ 1 and any k, if n + 1, ..., n + k are all\ncomposite, then there are distinct primes p₁, ... pₖ such that pᵢ ∣ n + i for all 1 ≤ i ≤ k.
\n Is Erdos375Prop true?
\n If Erdos375Prop is true, then (n + 1).nth Prime - n.nth Prime < (n.nth Prime) ^ (1 / 2 - c)\nfor some c > 0.
\n In particular, if Erdos375Prop is true, then Legendre's conjecture is asymptotically true.
\n It is easy to see that for any n ≥ 1 and k ≤ 2, if n + 1, ..., n + k are all composite,\nthen there are distinct primes p₁, ... pₖ such that pᵢ ∣ n + i for all 1 ≤ i ≤ k.
\n There exists a constant c > 0 such that for all n, if\nk < c * (log n / (log (log n))) ^ 3 → (∀ i < k, ¬ (n + i + 1).Prime), then\nthere are distinct primes p₁, ... pₖ such that pᵢ ∣ n + i for all 1 ≤ i ≤ k. This is proved\nin [RST75]. There is no need to only consider sufficiently large n because one can always take\nc small enough so that k < c * (log n / (log (log n))) ^ 3 implies that k = 0 until n is\nlarge.
\n Let $\\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\\subseteq {1,\\ldots,N}$ has\nsize at least $\\epsilon N$ then there must be distinct $a,b,c\\in A$ such that\n$$[a, b]=[b, c]=[a, c],$$\nwhere $[\\cdot, \\cdot]$ denotes the least common multiple?
"},"Erdos396.erdos_396":{"url":"/FormalConjectures/ErdosProblems/«396»/#Erdos396___erdos_396","anchor":"Erdos396___erdos_396","docHtml":"\n Is it true that for every $k$ there exists $n$ such that\n$$\\prod_{0\\leq i\\leq k}(n-i) \\mid \\binom{2n}{n}?$$
"},"Erdos370.erdos_370":{"url":"/FormalConjectures/ErdosProblems/«370»/#Erdos370___erdos_370","anchor":"Erdos370___erdos_370","docHtml":"\n Are there infinitely many $n$ such that the largest prime factor of $n$ is $< n^{\\frac{1}{2}}$ and\nthe largest prime factor of $n + 1$ is $< (n + 1)^{\\frac{1}{2}}$.
\n\n Steinerberger has pointed out this problem has a trivial solution.
\n\n This was formalized in Lean by Alexeev using Aristotle.
"},"Erdos520.IsRademacherMultiplicative":{"url":"/FormalConjectures/ErdosProblems/«520»/#Erdos520___IsRademacherMultiplicative","anchor":"Erdos520___IsRademacherMultiplicative","docHtml":"\n A random function $f$ is Rademacher multiplicative if $f(1) = 1$,\nfor each prime $p$, we independently choose $f(p) \\in {-1, 1}$ uniformly at random,\nfor each square-free integer $n = p_1 \\cdots p_r$, $f(n) = f(p_1) \\cdots f(p_r)$, and\nfor each non-squarefree integer $n$, $f(n) = 0$.
"},"Erdos520.IsRademacherMultiplicative.iIndepFun_primes":{"url":"/FormalConjectures/ErdosProblems/«520»/#Erdos520___IsRademacherMultiplicative___iIndepFun_primes","anchor":"Erdos520___IsRademacherMultiplicative___iIndepFun_primes","docHtml":"\n Prime entries are independent.
"},"Erdos520.IsRademacherMultiplicative.prob_of_prime":{"url":"/FormalConjectures/ErdosProblems/«520»/#Erdos520___IsRademacherMultiplicative___prob_of_prime","anchor":"Erdos520___IsRademacherMultiplicative___prob_of_prime","docHtml":"\n Primes entries are uniformly distributed on {-1, 1}.
\n Primes entries are uniformly distributed on {-1, 1}.
\n Primes entries are uniformly distributed on {-1, 1}.
\n Primes entries are uniformly distributed on {-1, 1}.
\n Let $f$ be a Rademacher multiplicative function.\nDoes there exist some constant $c > 0$ such that, almost surely,\n$$\n\\limsup_{N \\to \\infty} \\frac{\\sum_{m \\leq N} f(m)}{\\sqrt{N \\log \\log N}} = c?\n$$
"},"Erdos307.erdos_307":{"url":"/FormalConjectures/ErdosProblems/«307»/#Erdos307___erdos_307","anchor":"Erdos307___erdos_307","docHtml":"\n Are there two finite set of primes $P$ and $Q$ such that
\n\n $$\n1 = \\left( \\sum_{p \\in P} \\frac{1}{p} \\right) \\left( \\sum_{q \\in Q} \\frac{1}{q} \\right)\n$$\n?
\n\n Asked by Barbeau [Ba76].
\n\n [Ba76] Barbeau, E. J.,
\n Instead of asking for sets of primes, ask only that all elements in the sets be relatively coprime.
\n\n Cambie has found several examples when this weakened version is true. For example,\n$$\n1=\\left(1+\\frac{1}{5}\\right)\\left(\\frac{1}{2}+\\frac{1}{3}\\right)\n$$\nand\n$$\n1=\\left(1+\\frac{1}{41}\\right)\\left(\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{7}\\right).\n$$
"},"Erdos307.erdos_307.variants.coprime_one_notMem":{"url":"/FormalConjectures/ErdosProblems/«307»/#Erdos307___erdos_307___variants___coprime_one_notMem","anchor":"Erdos307___erdos_307___variants___coprime_one_notMem","docHtml":"\n There are no examples known of the weakened coprime version if we insist that $1\\not\\in P\\cup Q$.
"},"Erdos319.erdos_319":{"url":"/FormalConjectures/ErdosProblems/«319»/#Erdos319___erdos_319","anchor":"Erdos319___erdos_319","docHtml":"\n What is the size of the largest $A\\subseteq{1, \\dots, N}$ such that there is a function\n$\\delta : A \\to {-1, 1}$ such that\n$$\n\\sum_{n\\in A} \\frac{\\delta n}{n} = 0\n$$\nand\n$$\n\\sum_{n\\in A'}\\frac{\\delta n}{n} \\neq 0\n$$\nfor all non-empty $A'\\subsetneq A$.
"},"Erdos319.erdos_319.variants.isTheta":{"url":"/FormalConjectures/ErdosProblems/«319»/#Erdos319___erdos_319___variants___isTheta","anchor":"Erdos319___erdos_319___variants___isTheta","docHtml":"\n Let $c(N)$ be the size of the largest $A\\subseteq{1, \\dots, N}$ such that there is a function\n$\\delta : A \\to {-1, 1}$ such that\n$$\n\\sum_{n\\in A} \\frac{\\delta n}{n} = 0\n$$\nand\n$$\n\\sum_{n\\in A'}\\frac{\\delta n}{n} \\neq 0\n$$\nfor all non-empty $A'\\subsetneq A$. What is $\\Theta(c(N))$?
"},"Erdos319.erdos_319.variants.isBigO":{"url":"/FormalConjectures/ErdosProblems/«319»/#Erdos319___erdos_319___variants___isBigO","anchor":"Erdos319___erdos_319___variants___isBigO","docHtml":"\n Let $c(N)$ be the size of the largest $A\\subseteq{1, \\dots, N}$ such that there is a function\n$\\delta : A \\to {-1, 1}$ such that\n$$\n\\sum_{n\\in A} \\frac{\\delta n}{n} = 0\n$$\nand\n$$\n\\sum_{n\\in A'}\\frac{\\delta n}{n} \\neq 0\n$$\nfor all non-empty $A'\\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = O(g(N)).
"},"Erdos319.erdos_319.variants.isLittleO":{"url":"/FormalConjectures/ErdosProblems/«319»/#Erdos319___erdos_319___variants___isLittleO","anchor":"Erdos319___erdos_319___variants___isLittleO","docHtml":"\n Let $c(N)$ be the size of the largest $A\\subseteq{1, \\dots, N}$ such that there is a function\n$\\delta : A \\to {-1, 1}$ such that\n$$\n\\sum_{n\\in A} \\frac{\\delta n}{n} = 0\n$$\nand\n$$\n\\sum_{n\\in A'}\\frac{\\delta n}{n} \\neq 0\n$$\nfor all non-empty $A'\\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = o(g(N)).
"},"Erdos319.erdos_319.variants.lb":{"url":"/FormalConjectures/ErdosProblems/«319»/#Erdos319___erdos_319___variants___lb","anchor":"Erdos319___erdos_319___variants___lb","docHtml":"\n Adenwalla has observed that a lower bound (on the maximum size of $A$) of\n$$\n|A| \\geq (1 - \\frac{1}{e} + o(1))N\n$$\nfollows from the main result of Croot [Cr01].
\n\n [Cr01] Croot, III, Ernest S.,
\n The set of all possible numbers of unit distances determined by the vertices of a convex\n$n$-gon.
"},"Erdos96.convexUnitDistanceCounts_bddAbove":{"url":"/FormalConjectures/ErdosProblems/«96»/#Erdos96___convexUnitDistanceCounts_bddAbove","anchor":"Erdos96___convexUnitDistanceCounts_bddAbove","docHtml":"\n This lemma confirms that the set of possible unit-distance counts is bounded above, which\nensures that taking the supremum (sSup) is a well-defined operation. The trivial upper bound is\nthe total number of pairs of points, $\\binom{n}{2}$.
\n The maximum number of unit distances determined by the vertices of a convex $n$-gon.\nThis function is often denoted as $U_c(n)$ in combinatorics.
"},"Erdos96.erdos_96":{"url":"/FormalConjectures/ErdosProblems/«96»/#Erdos96___erdos_96","anchor":"Erdos96___erdos_96","docHtml":"\n If $n$ points in $\\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are\ndistance $1$ apart.
"},"Erdos507.minTriangleArea":{"url":"/FormalConjectures/ErdosProblems/«507»/#Erdos507___minTriangleArea","anchor":"Erdos507___minTriangleArea","docHtml":"\n The minimum area of a triangle determined by three distinct points in a set S.
\n $\\alpha(n)$ is the supremum of minTriangleArea S over all sets S of $n$ points in the unit disk.
\n Current best lower bound [KPS82].
"},"Erdos507.upperBarrier":{"url":"/FormalConjectures/ErdosProblems/«507»/#Erdos507___upperBarrier","anchor":"Erdos507___upperBarrier","docHtml":"\n The \"Barrier\" function: n^(-7/6) used for the best upper bound [CPZ24].
"},"Erdos507.erdos_507.equivalent":{"url":"/FormalConjectures/ErdosProblems/«507»/#Erdos507___erdos_507___equivalent","anchor":"Erdos507___erdos_507___equivalent","docHtml":"\n Let $\\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which\ndetermine a triangle of area at most $\\alpha(n)$. Estimate $\\alpha(n)$.
"},"Erdos507.erdos_507.lower":{"url":"/FormalConjectures/ErdosProblems/«507»/#Erdos507___erdos_507___lower","anchor":"Erdos507___erdos_507___lower","docHtml":"\n Estimate a lower bound for$\\alpha(n)$.
"},"Erdos507.erdos_507.upper":{"url":"/FormalConjectures/ErdosProblems/«507»/#Erdos507___erdos_507___upper","anchor":"Erdos507___erdos_507___upper","docHtml":"\n Estimate an upper bound for$\\alpha(n)$.
"},"Erdos507.erdos_507.variants.upper_trivial":{"url":"/FormalConjectures/ErdosProblems/«507»/#Erdos507___erdos_507___variants___upper_trivial","anchor":"Erdos507___erdos_507___variants___upper_trivial","docHtml":"\n It is trivial that $\\alpha(n) \\ll 1/n$.
"},"Erdos507.erdos_507.variants.lower_erdos":{"url":"/FormalConjectures/ErdosProblems/«507»/#Erdos507___erdos_507___variants___lower_erdos","anchor":"Erdos507___erdos_507___variants___lower_erdos","docHtml":"\n Erdős observed that $\\alpha(n) \\gg 1/n^2$.
"},"Erdos507.erdos_507.variants.lower_kps82":{"url":"/FormalConjectures/ErdosProblems/«507»/#Erdos507___erdos_507___variants___lower_kps82","anchor":"Erdos507___erdos_507___variants___lower_kps82","docHtml":"\n Current best lower bound [KPS82].
"},"Erdos507.erdos_507.variants.upper_cpz24":{"url":"/FormalConjectures/ErdosProblems/«507»/#Erdos507___erdos_507___variants___upper_cpz24","anchor":"Erdos507___erdos_507___variants___upper_cpz24","docHtml":"\n Current best upper bound [CPZ24]: $\\alpha(n) \\ll n^{-7/6 + o(1)}$.
"},"Erdos10.sumPrimeAndTwoPows":{"url":"/FormalConjectures/ErdosProblems/«10»/#Erdos10___sumPrimeAndTwoPows","anchor":"Erdos10___sumPrimeAndTwoPows","docHtml":"\n The set of natural numbers that can be written as a sum\nof a prime and at most $k$ powers of $2$.
"},"Erdos10.erdos_10":{"url":"/FormalConjectures/ErdosProblems/«10»/#Erdos10___erdos_10","anchor":"Erdos10___erdos_10","docHtml":"\n Is there some $k$ such that every integer is the sum of a prime and at most $k$\npowers of $2$?
"},"Erdos10.erdos_10.variants.gallagher":{"url":"/FormalConjectures/ErdosProblems/«10»/#Erdos10___erdos_10___variants___gallagher","anchor":"Erdos10___erdos_10___variants___gallagher","docHtml":"\n Gallagher [Ga75] has shown that for any $ϵ > 0$ there exists $k(ϵ)$\nsuch that the set of integers which are the sum of a prime and at most $k(ϵ)$\nmany powers of $2$ has lower density at least $1 - ϵ$.
\n\n Ref: Gallagher, P. X.,
\n Granville and Soundararajan [GrSo98] have conjectured that at most $3$\npowers of $2$ suffice for all odd integers, and hence at most $4$ powers of $2$\nsuffice for all even integers.
\n\n Ref: Granville, A. and Soundararajan, K.,
\n Bogdan Grechuk has observed that 1117175146 is not the sum of a prime\nand at most $3$ powers of $2$.
\n There are infinitely many even integers not the sum of a prime and $2$ powers of $2$
"},"Erdos10.erdos_10.variants.grechuk":{"url":"/FormalConjectures/ErdosProblems/«10»/#Erdos10___erdos_10___variants___grechuk","anchor":"Erdos10___erdos_10___variants___grechuk","docHtml":"\n Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$\npowers of $2$, and pointed out that parity considerations, coupled with the fact that there\nare many integers not the sum of a prime and $2$ powers of $2$ suggest that there exist\ninfinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).
"},"Erdos18.practicalH":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___practicalH","anchor":"Erdos18___practicalH","docHtml":"\n For a practical number $n$, $h(n)$ is the maximum over all $1 ≤ m ≤ n$ of\nthe minimum number of divisors of $n$ needed to represent $m$ as a sum of\ndistinct divisors.
"},"Erdos18.practicalH_one":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___practicalH_one","anchor":"Erdos18___practicalH_one","docHtml":"\n $h(1) = 1$: we need the single divisor {1} to represent 1.
"},"Erdos18.practicalH_two":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___practicalH_two","anchor":"Erdos18___practicalH_two","docHtml":"\n $h(2) = 1$: divisors are {1, 2}, each of m=1,2 needs only 1 divisor.
"},"Erdos18.practicalH_six":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___practicalH_six","anchor":"Erdos18___practicalH_six","docHtml":"\n $h(6) = 2$: divisors are {1, 2, 3, 6}. The hardest m to represent is\nm=4 or m=5, each requiring 2 divisors: 4=1+3, 5=2+3.
"},"Erdos18.practicalH_twelve":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___practicalH_twelve","anchor":"Erdos18___practicalH_twelve","docHtml":"\n $h(12) = 3$: divisors are {1, 2, 3, 4, 6, 12}. The hardest m is\nm=11, requiring 3 divisors: 11=1+4+6.
"},"Erdos18.practicalH_le_divisors":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___practicalH_le_divisors","anchor":"Erdos18___practicalH_le_divisors","docHtml":"\n For any practical number $n$, $h(n) ≤ number of divisors of $n$.
"},"Erdos18.factorial_isPractical":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___factorial_isPractical","anchor":"Erdos18___factorial_isPractical","docHtml":"\n $h(n!)$ is well-defined since $n!$ is practical for $n ≥ 1$.
"},"Erdos18.erdos_18a":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___erdos_18a","anchor":"Erdos18___erdos_18a","docHtml":"\nConjecture 1.\nAre there infinitely many practical numbers $m$ such that $h(m) < (\\log \\log m)^{O(1)}$?
\n\n More precisely: does there exist a constant $C > 0$ such that for infinitely many\npractical numbers $m$, we have $h(m) < (\\log \\log m)^C$?
"},"Erdos18.erdos_18b":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___erdos_18b","anchor":"Erdos18___erdos_18b","docHtml":"\nConjecture 2.\nIs it true that $h(n!) < n^{o(1)}$? That is, for all $\\varepsilon > 0$,\nis $h(n!) < n^\\varepsilon$ for sufficiently large $n$?
"},"Erdos18.erdos_18c":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___erdos_18c","anchor":"Erdos18___erdos_18c","docHtml":"\nConjecture 3.\nOr perhaps even $h(n!) < (\\log n)^{O(1)}$?
\n\n Erdős offered $250 for a proof or disproof.
"},"Erdos18.erdos_18_upper_bound":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___erdos_18_upper_bound","anchor":"Erdos18___erdos_18_upper_bound","docHtml":"\nErdős's Theorem.\nErdős proved that $h(n!) < n$ for all $n \\ge 1$.
"},"Erdos18.erdos_18_vose":{"url":"/FormalConjectures/ErdosProblems/«18»/#Erdos18___erdos_18_vose","anchor":"Erdos18___erdos_18_vose","docHtml":"\nVose's Theorem.\nVose proved the existence of infinitely many practical numbers $m$ such that\n$h(m) \\ll (\\log m)^{1/2}$. This gives a positive answer to a weaker form of Conjecture 1.
"},"Erdos243.erdos_243":{"url":"/FormalConjectures/ErdosProblems/«243»/#Erdos243___erdos_243","anchor":"Erdos243___erdos_243","docHtml":"\n Let $a_1 < a_2 < \\dots$ be a sequence of integers such that\n$\\lim_{n\\to\\infty} \\frac{a_n}{a_{n-1}^2} = 1$ and $\\sum \\frac{1}{a_n} \\in \\mathbb{Q}$.
\n\n Then, for all sufficiently large $n \\ge 1$, $a_n = a_{n-1}^2 - a_{n-1} + 1$.
"},"Erdos91.IsOptimal":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___IsOptimal","anchor":"Erdos91___IsOptimal","docHtml":"\n A set $A$ is 'optimal' if it has $n$ points and achieves the minimum distance count.
"},"Erdos91.DilationEquivSimilar":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___DilationEquivSimilar","anchor":"Erdos91___DilationEquivSimilar","docHtml":"\n Two finite sets of points in $\\mathbb{R}^2$ are similar if one can be mapped to the other by a\nDilationEquiv.
"},"Erdos91.equiTriangle":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___equiTriangle","anchor":"Erdos91___equiTriangle","docHtml":"\n Equilateral triangle with unit side length, resting on the x-axis with one vertex at the origin.
"},"Erdos91.unitSquare":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___unitSquare","anchor":"Erdos91___unitSquare"},"Erdos91.circleSeven":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___circleSeven","anchor":"Erdos91___circleSeven","docHtml":"\n Regular 7-gon with unit side length, touching both axes in the first quadrant.
"},"Erdos91.wheelSeven":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___wheelSeven","anchor":"Erdos91___wheelSeven","docHtml":"\n Wheel graph on 7 vertices (center + regular hexagon) with unit side length,\ntouching both axes in the first quadrant.
"},"Erdos91.erdos_91.test.equiTriangle_optimal":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___test___equiTriangle_optimal","anchor":"Erdos91___erdos_91___test___equiTriangle_optimal"},"Erdos91.erdos_91.test.equiTriangle_unique_optimal":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___test___equiTriangle_unique_optimal","anchor":"Erdos91___erdos_91___test___equiTriangle_unique_optimal"},"Erdos91.erdos_91.test.unitSquare_optimal":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___test___unitSquare_optimal","anchor":"Erdos91___erdos_91___test___unitSquare_optimal"},"Erdos91.erdos_91.test.circleSeven_optimal":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___test___circleSeven_optimal","anchor":"Erdos91___erdos_91___test___circleSeven_optimal"},"Erdos91.erdos_91.test.wheelSeven_optimal":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___test___wheelSeven_optimal","anchor":"Erdos91___erdos_91___test___wheelSeven_optimal"},"Erdos91.erdos_91.test.dissimilar_circleSeven_wheelSeven":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___test___dissimilar_circleSeven_wheelSeven","anchor":"Erdos91___erdos_91___test___dissimilar_circleSeven_wheelSeven"},"Erdos91.UniqueMinimizer":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___UniqueMinimizer","anchor":"Erdos91___UniqueMinimizer","docHtml":"\n The predicate on $n$ asserting all $A, B\\subset \\mathbb{R}^2$,\nwith $\\lvert A\\rvert=n = \\lvert B\\rvert$, which minimise the number of distinct points for all sets\nwith $n$ elements are similar.
"},"Erdos91.erdos_91":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91","anchor":"Erdos91___erdos_91","docHtml":"\n Suppose $A\\subset \\mathbb{R}^2$ has $\\lvert A\\rvert=n$ and minimises the number of distinct\ndistances between points in $A$. Prove that for large $n$ there are at least two\n(and probably many) such $A$ which are non-similar.
"},"Erdos91.erdos_91.variants.three":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___variants___three","anchor":"Erdos91___erdos_91___variants___three","docHtml":"\n For $n = 3$ the equilateral triangle is the only such set.
"},"Erdos91.erdos_91.variants.four":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___variants___four","anchor":"Erdos91___erdos_91___variants___four","docHtml":"\n For $n=4$ the square or two equilateral triangles sharing an edge give two\nnon-similar examples.
"},"Erdos91.erdos_91.variants.five":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___variants___five","anchor":"Erdos91___erdos_91___variants___five","docHtml":"\n For $n = 5$ the regular pentagon is the unique such set (which has two distinct distances).\nErdős mysteriously remarks in [Er90] this was proved by 'a colleague'. (In [Er87b] this is\ndescribed as 'a colleague from Zagreb (unfortunately I do not have his letter)'.)\nA published proof of this fact is provided by Kovács [Ko24c].
"},"Erdos91.erdos_91.variants.six":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___variants___six","anchor":"Erdos91___erdos_91___variants___six","docHtml":"\n In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 6$.
"},"Erdos91.erdos_91.variants.seven":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___variants___seven","anchor":"Erdos91___erdos_91___variants___seven","docHtml":"\n In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 7$.
"},"Erdos91.erdos_91.variants.eight":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___variants___eight","anchor":"Erdos91___erdos_91___variants___eight","docHtml":"\n In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 8$.
"},"Erdos91.erdos_91.variants.nine":{"url":"/FormalConjectures/ErdosProblems/«91»/#Erdos91___erdos_91___variants___nine","anchor":"Erdos91___erdos_91___variants___nine","docHtml":"\n In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 9$.
"},"Erdos412.erdos_412":{"url":"/FormalConjectures/ErdosProblems/«412»/#Erdos412___erdos_412","anchor":"Erdos412___erdos_412","docHtml":"\n Let $σ_1(n)=σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$.\nIs it true that, for every $m, n ≥ 2$, there exist some $i, j$ such that $σ_i(m) = σ_j(n)$?
"},"Erdos677.lcmInterval_eq_example1":{"url":"/FormalConjectures/ErdosProblems/«677»/#Erdos677___lcmInterval_eq_example1","anchor":"Erdos677___lcmInterval_eq_example1","docHtml":"\n Erdős expected very few solutions for $M(n, k) = M(m, l)$, where $m \\geq n + k$ and $l > 1$.\nThe only solutions he knew were the following.
"},"Erdos677.erdos_677":{"url":"/FormalConjectures/ErdosProblems/«677»/#Erdos677___erdos_677","anchor":"Erdos677___erdos_677","docHtml":"\n Denote by $M(n, k)$ the least common multiple of the finite set ${n+1, \\dotsc, n+k}$.\nIs it true that for all $m \\geq n + k$, we get $M(m, k) \\neq M(n, k)$?
"},"Erdos770.h":{"url":"/FormalConjectures/ErdosProblems/«770»/#Erdos770___h","anchor":"Erdos770___h","docHtml":"\n Let $h n$ be the minimal number such that $2 ^ n - 1, \\dots, h(n) ^ n - 1$\nare collectively coprime.
"},"Erdos770.Nat.Prime.h_eq_add_one":{"url":"/FormalConjectures/ErdosProblems/«770»/#Erdos770___Nat___Prime___h_eq_add_one","anchor":"Erdos770___Nat___Prime___h_eq_add_one","docHtml":"\nn + 1 is prime iff h n = n + 1. This is described as 'easy to see' in [Er74b].
\n For odd n, the values of h n form an unbounded set.\nThis is described as 'easy to see' in [Er74b].
\n For every prime p, does the density of integers with h n = p exist?
\n Does liminf h n = ∞?
\n Is it true that if p is the greatest prime such that p - 1 ∣ n and p > n ^ ε, then\nh n = p?
\n It is probably true that h n = 3 for infinitely many n.
\n The set of integers not divisible by any u_i.
"},"Erdos1101.A":{"url":"/FormalConjectures/ErdosProblems/«1101»/#Erdos1101___A","anchor":"Erdos1101___A","docHtml":"\n The sequence of integers A_u which are not divisible by any u_i\narranged in a monotonic sequence.
"},"Erdos1101.t":{"url":"/FormalConjectures/ErdosProblems/«1101»/#Erdos1101___t","anchor":"Erdos1101___t","docHtml":"\n t_x such that u_0 ... u_{t_x-1} ≤ x < u_0 ... u_{t_x}.
"},"Erdos1101.IsGood":{"url":"/FormalConjectures/ErdosProblems/«1101»/#Erdos1101___IsGood","anchor":"Erdos1101___IsGood","docHtml":"\n A sequence is \"good\" if
\n\n it is strictly monotone
\n\n it is pairwise coprime
\n\n the sum of reciprocals converges
\n\n the gap between consecutive elements in A(u) is bounded relative to t_x.
\n\n There is NO good sequence with polynomial growth.
\n\n There is a good sequence with sub-exponential growth.
\n\n Erdős Problem 855 (Segal's conjecture): $\\pi(x + y) \\le \\pi(x) + \\pi(y)$\nfor sufficiently large $x, y$.
"},"Erdos503.erdos_503":{"url":"/FormalConjectures/ErdosProblems/«503»/#Erdos503___erdos_503","anchor":"Erdos503___erdos_503","docHtml":"\n What is the size of the largest $A \\subseteq \\mathbb{R}^n$ such that every three points from $A$\ndetermine an isosceles triangle? That is, for any three points $x$, $y$, $z$ from $A$, at least two\nof the distances $|x - y|$, $|y - z|$, $|x - z|$ are equal.
"},"Erdos503.erdos_503.variants.R2":{"url":"/FormalConjectures/ErdosProblems/«503»/#Erdos503___erdos_503___variants___R2","anchor":"Erdos503___erdos_503___variants___R2","docHtml":"\n When $n = 2$, the answer is 6 (due to Kelly [ErKe47] - an alternative proof is given by Kovács [Ko24c]).
\n\n [ErKe47] Erdős, Paul and Kelly, L. M., Elementary Problems and Solutions: Solutions: E735. Amer. Math. Monthly (1947), 227-229.\n[Ko24c] Z. Kovács, A note on Erdős's mysterious remark. arXiv:2412.05190 (2024).
"},"Erdos503.erdos_503.variants.R3":{"url":"/FormalConjectures/ErdosProblems/«503»/#Erdos503___erdos_503___variants___R3","anchor":"Erdos503___erdos_503___variants___R3","docHtml":"\n When $n = 3$, the answer is 8 (due to Croft [Cr62]).
\n\n [Cr62] Croft, H. T., $9$-point and $7$-point configurations in $3$-space. Proc. London Math. Soc. (3) (1962), 400-424.
"},"Erdos503.erdos_503.variants.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«503»/#Erdos503___erdos_503___variants___upper_bound","anchor":"Erdos503___erdos_503___variants___upper_bound","docHtml":"\n The best upper bound known in general is due to Blokhius [Bl84] who showed that\n$$\n|A| \\leq \\binom{n + 2}{2}\n$$
\n\n [Bl84] Blokhuis, A., Few-distance sets. (1984), iv+70.
"},"Erdos503.erdos_503.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«503»/#Erdos503___erdos_503___variants___lower_bound","anchor":"Erdos503___erdos_503___variants___lower_bound","docHtml":"\n Alweiss has observed a lower bound of $\\binom{n + 1}{2}$ follows from considering the subset of\n$\\mathbb{R}^{n + 1}$ formed of all vectors $e_i + e_j$ where $e_i$, $e_j$ are distinct coordinate\nvectors. This set can be viewed as a subset of some $\\mathbb{R}^n$, and is easily checked to have\nthe required property.
"},"Erdos566.erdos_566":{"url":"/FormalConjectures/ErdosProblems/«566»/#Erdos566___erdos_566","anchor":"Erdos566___erdos_566","docHtml":"\n Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges.\nIs it true that, if $H$ has $m$ edges and no isolated vertices, then $\\hat{r}(G,H) \\ll m$?
\n\n In other words: if $G$ is sparse (every induced subgraph on $k$ vertices has $≤ 2k-3$ edges),\nis $G$ Ramsey size linear?
"},"Erdos1085.f":{"url":"/FormalConjectures/ErdosProblems/«1085»/#Erdos1085___f","anchor":"Erdos1085___f","docHtml":"\n The maximal number of pairs of points which are distance 1 apart that a set of n points in\nℝ^d make.
\n Erdős showed $f_2(n) > n^{1+c/\\log\\log n}$ for some $c > 0$.
"},"Erdos1085.erdos_1085.variants.upper_d2":{"url":"/FormalConjectures/ErdosProblems/«1085»/#Erdos1085___erdos_1085___variants___upper_d2","anchor":"Erdos1085___erdos_1085___variants___upper_d2","docHtml":"\n Spencer, Szemerédi, and Trotter showed $f_2(n) = O(n^{4/3})$.
"},"Erdos1085.erdos_1085.variants.lower_d3":{"url":"/FormalConjectures/ErdosProblems/«1085»/#Erdos1085___erdos_1085___variants___lower_d3","anchor":"Erdos1085___erdos_1085___variants___lower_d3","docHtml":"\n Erdős showed $f_3(n) = Ω(n^{4/3}\\log\\log n)$.
"},"Erdos1085.erdos_1085.variants.upper_d3":{"url":"/FormalConjectures/ErdosProblems/«1085»/#Erdos1085___erdos_1085___variants___upper_d3","anchor":"Erdos1085___erdos_1085___variants___upper_d3","docHtml":"\n Is the $n^{4/3}\\log\\log n$ lower bound in 3D also an upper bound?.
"},"Erdos1085.erdos_1085.variants.lower_d4_lenz":{"url":"/FormalConjectures/ErdosProblems/«1085»/#Erdos1085___erdos_1085___variants___lower_d4_lenz","anchor":"Erdos1085___erdos_1085___variants___lower_d4_lenz","docHtml":"\n Lenz showed that, for $d \\ge 4$, $f_d(n) \\ge \\frac{p - 1}{2p} n^2 - O(1)$ where\n$p = \\lfloor\\frac d2\\rfloor$.
"},"Erdos1085.erdos_1085.variants.upper_d4_erdos":{"url":"/FormalConjectures/ErdosProblems/«1085»/#Erdos1085___erdos_1085___variants___upper_d4_erdos","anchor":"Erdos1085___erdos_1085___variants___upper_d4_erdos","docHtml":"\n Erdős showed that, for $d \\ge 4$, $f_d(n) \\le \\left(\\frac{p - 1}{2p} + o(1)\\right) n^2$ where\n$p = \\lfloor\\frac d2\\rfloor$.
"},"Erdos1085.erdos_1085.variants.upper_lower_d5_odd":{"url":"/FormalConjectures/ErdosProblems/«1085»/#Erdos1085___erdos_1085___variants___upper_lower_d5_odd","anchor":"Erdos1085___erdos_1085___variants___upper_lower_d5_odd","docHtml":"\n Erdős and Pach showed that, for $d \\ge 5$ odd, there exist constants $c_1(d), c_2(d) > 0$\nsuch that $\\frac{p - 1}{2p} n^2 - c_1 n^{4/3} ≤ f_d(n) \\le \\frac{p - 1}{2p} n^2 + c_2 n^{4/3}$ where\n$p = \\lfloor\\frac d2\\rfloor$.
"},"Erdos1094.erdos_1094":{"url":"/FormalConjectures/ErdosProblems/«1094»/#Erdos1094___erdos_1094","anchor":"Erdos1094___erdos_1094","docHtml":"\n For all $n\\ge 2k$ the least prime factor of $\\binom{n}{k}$ is $\\le\\max(n/k,k)$, with only\nfinitely many exceptions.
"},"Erdos1082.erdos_1082.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1082»/#Erdos1082___erdos_1082___parts___i","anchor":"Erdos1082___erdos_1082___parts___i","docHtml":"\n Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points with no three on a line.\nDoes $A$ determine at least $\\lfloor n/2\\rfloor$ distinct distances?
"},"Erdos1082.erdos_1082.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«1082»/#Erdos1082___erdos_1082___parts___ii","anchor":"Erdos1082___erdos_1082___parts___ii","docHtml":"\n Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points with no three on a line.\nMust there exist a single point from which there are at least $\\lfloor n/2\\rfloor$ distinct\ndistances?
\n\n This question has been answered negatively by Xichuan in the\ncomments, who gave a set of $42$ points in\n$\\mathbb{R}^2$, with no three on a line, such that each point determines only $20$ distinct distances.
\n\n A smaller counterexample has been formalised here: it comprised of $8$ points, where each point only\ndetermines $3$ distances.
\n\n This counterexample has originally been found by Heiko Harborth.
"},"Erdos288.erdos_288":{"url":"/FormalConjectures/ErdosProblems/«288»/#Erdos288___erdos_288","anchor":"Erdos288___erdos_288","docHtml":"\n Is it true that there are only finitely many pairs of intervals $I_1$, $I_2$ such that\n$$\n\\sum_{n_1 \\in I_1} \\frac{1}{n_1} + \\sum_{n_2 \\in I_2} \\frac{1}{n_2} \\in \\mathbb{N}?\n$$
"},"Erdos288.erdos_288.variants.i2_card_eq_1":{"url":"/FormalConjectures/ErdosProblems/«288»/#Erdos288___erdos_288___variants___i2_card_eq_1","anchor":"Erdos288___erdos_288___variants___i2_card_eq_1","docHtml":"\n This is still open even if $|I_2| = 1$.
"},"Erdos288.erdos_288.variants.k_intervals":{"url":"/FormalConjectures/ErdosProblems/«288»/#Erdos288___erdos_288___variants___k_intervals","anchor":"Erdos288___erdos_288___variants___k_intervals","docHtml":"\n It is perhaps true with two intervals replaced by any $k$ intervals.
"},"Erdos288.erdos_288.variants.exists_k_gt_2":{"url":"/FormalConjectures/ErdosProblems/«288»/#Erdos288___erdos_288___variants___exists_k_gt_2","anchor":"Erdos288___erdos_288___variants___exists_k_gt_2","docHtml":"\n Is it true for any $k > 2$ that only finitely many $k$ intervals satisfy this condition?
"},"Erdos279.erdos_279":{"url":"/FormalConjectures/ErdosProblems/«279»/#Erdos279___erdos_279","anchor":"Erdos279___erdos_279","docHtml":"\n Let $k\\geq 3$. Is there a choice of congruence classes $a_p\\pmod{p}$ for every prime $p$\nsuch that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$\nand integer $t\\geq k$?
"},"Erdos1061.S":{"url":"/FormalConjectures/ErdosProblems/«1061»/#Erdos1061___S","anchor":"Erdos1061___S","docHtml":"\n Let S x count the number of ordered pairs of positive integers (a, b) with a + b ≤ x\nsuch that σ(a) + σ(b) = σ(a + b), where σ is the sum of divisors function.
\n In particular, (a, b) and (b, a) are counted separately; an unordered variant could be obtained\nby additionally requiring a ≤ b.
\n How many (ordered) solutions are there to σ(a) + σ(b) = σ(a + b) with a + b ≤ x?\nIs it true that this number is asymptotic to c * x for some constant c > 0?
\n The property that $n > 2$ and $n - 2^k$ is prime for all $k \\geq 1$ with $2^k < n$.
\n\n Following the OEIS A039669 convention (\"Numbers n > 2 such that ...\"),\nwe require $n > 2$ to exclude the trivial cases $n \\leq 2$, for which the primality condition\nis vacuously satisfied.
"},"Erdos1142.erdos_1142":{"url":"/FormalConjectures/ErdosProblems/«1142»/#Erdos1142___erdos_1142","anchor":"Erdos1142___erdos_1142","docHtml":"\n Are there infinitely many $n > 2$ such that $n - 2^k$ is prime for all $k \\geq 1$ with $2^k < n$?
\n\n The only known such $n$ are $4, 7, 15, 21, 45, 75, 105$ (OEIS A039669).
"},"Erdos1142.erdos_1142.variants.mientka_weitzenkamp":{"url":"/FormalConjectures/ErdosProblems/«1142»/#Erdos1142___erdos_1142___variants___mientka_weitzenkamp","anchor":"Erdos1142___erdos_1142___variants___mientka_weitzenkamp","docHtml":"\n Mientka and Weitzenkamp [MiWe69] proved that the only $n \\leq 2^{44}$ such that $n > 2$ and\n$n - 2^k$ is prime for all $k \\geq 1$ with $2^k < n$ are $4, 7, 15, 21, 45, 75, 105$.
"},"Erdos1142.erdos_1142.test_4":{"url":"/FormalConjectures/ErdosProblems/«1142»/#Erdos1142___erdos_1142___test_4","anchor":"Erdos1142___erdos_1142___test_4","docHtml":"\n $4$ satisfies the Erdős 1142 property: $4 - 2 = 2$ is prime.
"},"Erdos1142.erdos_1142.test_7":{"url":"/FormalConjectures/ErdosProblems/«1142»/#Erdos1142___erdos_1142___test_7","anchor":"Erdos1142___erdos_1142___test_7","docHtml":"\n $7$ satisfies the Erdős 1142 property: $7 - 2 = 5$ and $7 - 4 = 3$ are prime.
"},"Erdos1142.erdos_1142.test_15":{"url":"/FormalConjectures/ErdosProblems/«1142»/#Erdos1142___erdos_1142___test_15","anchor":"Erdos1142___erdos_1142___test_15","docHtml":"\n $15$ satisfies the Erdős 1142 property: $15 - 2 = 13$, $15 - 4 = 11$, $15 - 8 = 7$.
"},"Erdos1142.erdos_1142.test_21":{"url":"/FormalConjectures/ErdosProblems/«1142»/#Erdos1142___erdos_1142___test_21","anchor":"Erdos1142___erdos_1142___test_21","docHtml":"\n $21$ satisfies the Erdős 1142 property: $21 - 2 = 19$, $21 - 4 = 17$, $21 - 8 = 13$,\n$21 - 16 = 5$.
"},"Erdos1142.erdos_1142.test_45":{"url":"/FormalConjectures/ErdosProblems/«1142»/#Erdos1142___erdos_1142___test_45","anchor":"Erdos1142___erdos_1142___test_45","docHtml":"\n $45$ satisfies the Erdős 1142 property: $45 - 2 = 43$, $45 - 4 = 41$, $45 - 8 = 37$,\n$45 - 16 = 29$, $45 - 32 = 13$.
"},"Erdos1142.erdos_1142.test_75":{"url":"/FormalConjectures/ErdosProblems/«1142»/#Erdos1142___erdos_1142___test_75","anchor":"Erdos1142___erdos_1142___test_75","docHtml":"\n $75$ satisfies the Erdős 1142 property: $75 - 2 = 73$, $75 - 4 = 71$, $75 - 8 = 67$,\n$75 - 16 = 59$, $75 - 32 = 43$, $75 - 64 = 11$.
"},"Erdos1142.erdos_1142.test_105":{"url":"/FormalConjectures/ErdosProblems/«1142»/#Erdos1142___erdos_1142___test_105","anchor":"Erdos1142___erdos_1142___test_105","docHtml":"\n $105$ satisfies the Erdős 1142 property: the largest known example.\n$105 - 2 = 103$, $105 - 4 = 101$, $105 - 8 = 97$, $105 - 16 = 89$, $105 - 32 = 73$,\n$105 - 64 = 41$.
"},"Erdos1142.erdos_1142.test_not_106":{"url":"/FormalConjectures/ErdosProblems/«1142»/#Erdos1142___erdos_1142___test_not_106","anchor":"Erdos1142___erdos_1142___test_not_106","docHtml":"\n $106$ does not satisfy the Erdős 1142 property ($106 - 2 = 104 = 8 \\times 13$).
"},"Erdos886.Erdos886Divisors":{"url":"/FormalConjectures/ErdosProblems/«886»/#Erdos886___Erdos886Divisors","anchor":"Erdos886___Erdos886Divisors","docHtml":"\n The set of divisors of $n$ in the interval $(n^{1/2}, n^{1/2} + n^{1/2-\\epsilon})$.
"},"Erdos886.erdos_886":{"url":"/FormalConjectures/ErdosProblems/«886»/#Erdos886___erdos_886","anchor":"Erdos886___erdos_886","docHtml":"\n Let $\\epsilon>0$. Is it true that, for all large $n$, the number of divisors of $n$ in\n$(n^{1/2},n^{1/2}+n^{1/2-\\epsilon})$ is $O_\\epsilon(1)$?
\n\n Erdős attributes this conjecture to Ruzsa.
"},"Erdos886.erdos_886.variants.rosenfeld_infinite":{"url":"/FormalConjectures/ErdosProblems/«886»/#Erdos886___erdos_886___variants___rosenfeld_infinite","anchor":"Erdos886___erdos_886___variants___rosenfeld_infinite","docHtml":"\n Erdős and Rosenfeld [ErRo97] proved that there are infinitely many $n$ such that there are\nfour divisors of $n$ in $(n^{1/2},n^{1/2}+16n^{1/4})$.
"},"Erdos886.erdos_886.variants.rosenfeld_bound":{"url":"/FormalConjectures/ErdosProblems/«886»/#Erdos886___erdos_886___variants___rosenfeld_bound","anchor":"Erdos886___erdos_886___variants___rosenfeld_bound","docHtml":"\n Erdős and Rosenfeld [ErRo97] proved that, for any constant $C>0$, all large $n$ have at most\n$1+C^2$ many divisors in $[n^{1/2}, n^{1/2}+Cn^{1/4}]$.
"},"Erdos379.S":{"url":"/FormalConjectures/ErdosProblems/«379»/#Erdos379___S","anchor":"Erdos379___S","docHtml":"\n
\n Let $S(n)$ denote the largest integer such that, for all $1 ≤ k < n$, the binomial coefficient\n$\\binom{n}{k}$ is divisible by $p^S(n)$ for some prime $p$ (depending on $k$).Then\n$\\limsup S(n) = \\infty$.
\n\n This was formalized in Lean by Tao.
"},"Erdos332.D_A":{"url":"/FormalConjectures/ErdosProblems/«332»/#Erdos332___D_A","anchor":"Erdos332___D_A","docHtml":"\n The set of numbers $D(A)$ which occur infinitely often as $a_1 - a_2$ with $a_1, a_2 \\in A$.
"},"Erdos332.HasBoundedGaps":{"url":"/FormalConjectures/ErdosProblems/«332»/#Erdos332___HasBoundedGaps","anchor":"Erdos332___HasBoundedGaps","docHtml":"\n A set $S \\subseteq \\mathbb{Z}$ has bounded gaps if it is syndetic, meaning there is a uniform\nbound $M$ such that every interval of length $M$ contains an element of $S$.
"},"Erdos332.erdos_332":{"url":"/FormalConjectures/ErdosProblems/«332»/#Erdos332___erdos_332","anchor":"Erdos332___erdos_332","docHtml":"\n Let $A\\subseteq \\mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as\n$a_1 - a_2$ with $a_1, a_2\\in A$. What conditions on $A$ are sufficient to ensure $D(A)$ has bounded\ngaps?
\n\n This is formalised here using the answer(sorry) mechanism. In order to solve this problem one\nhas to provide what the sufficient conditions are, and proof that they imply the desired condition.\nIf the condition is a solution to the problem is up to human judgement.
\n For each $m$ and $\\alpha$, the density of the set of integers which are divisible by\nsome $d \\equiv 1 \\pmod{m}$ with $1 < d < \\exp (m ^ \\alpha)$ exists.
"},"Erdos697.δ":{"url":"/FormalConjectures/ErdosProblems/«697»/#Erdos697______","anchor":"Erdos697______","docHtml":"\n For each $m$ and $\\alpha$, $\\delta (m, \\alpha)$ is the density of the set of integers which are\ndivisible by some $d \\equiv 1 \\pmod{m}$ with $1 < d < exp (m ^ \\alpha)$ exists.
"},"Erdos697.erdos_697.variants.delta_lt":{"url":"/FormalConjectures/ErdosProblems/«697»/#Erdos697___erdos_697___variants___delta_lt","anchor":"Erdos697___erdos_697___variants___delta_lt","docHtml":"\n $\\delta < \\frac{m ^ \\alpha + 1}{m}`. This shows that\n$lim_{m\\rightarrow\\infty} \\delta (m, \\alpha) = 0$ for $\\alpha < 1$.\n#TODO: prove this theorem.
"},"Erdos697.erdos_697.parts.i":{"url":"/FormalConjectures/ErdosProblems/«697»/#Erdos697___erdos_697___parts___i","anchor":"Erdos697___erdos_697___parts___i","docHtml":"\n Let $\\beta = \\frac{1}{\\log 2}$. Then $lim_{m\\rightarrow\\infty} \\delta (m, \\alpha) = 0$ if\n$\\alpha < \\beta$. This is proved in [Ha92].
"},"Erdos697.erdos_697.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«697»/#Erdos697___erdos_697___parts___ii","anchor":"Erdos697___erdos_697___parts___ii","docHtml":"\n $lim_{m\\rightarrow\\infty} \\delta (m, \\alpha) = 1$ if $\\beta < \\alpha$.\nThis is proved in [Ha92].
"},"Erdos198.baumgartner_strong":{"url":"/FormalConjectures/ErdosProblems/«198»/#Erdos198___baumgartner_strong","anchor":"Erdos198___baumgartner_strong","docHtml":"\n Let $V$ be a vector space over the rationals and let $k$ be a fixed\npositive integer. Then there is a set $X_k \\subseteq V$ such that $X_k$ meets\nevery infinite arithmetic progression in $V$ but $X_k$ intersects every\n$k$-element arithmetic progression in at most two points.
\n\n At the end of [Ba75] the author claims that by \"slightly modifying the method of [his proof]\", one\ncan prove this.
"},"Erdos198.baumgartner_headline":{"url":"/FormalConjectures/ErdosProblems/«198»/#Erdos198___baumgartner_headline","anchor":"Erdos198___baumgartner_headline","docHtml":"\n The statement for which Baumgartner actually writes a proof.
"},"Erdos198.erdos_198":{"url":"/FormalConjectures/ErdosProblems/«198»/#Erdos198___erdos_198","anchor":"Erdos198___erdos_198","docHtml":"\n The answer is no; Erdős and Graham report this was proved by Baumgartner, presumably referring to\nthe paper [Ba75], which does not state this exactly, but the following simple construction is\nimplicit in [Ba75].
\n\n Let $P_1,P_2,\\ldots$ be an enumeration of all countably many infinite arithmetic progressions. We\nchoose $a_1$ to be the minimal element of $P_1\\cap \\mathbb{N}$, and in general choose $a_n$ to be an\nelement of $P_n\\cap \\mathbb{N}$ such that $a_n>2a_{n-1}$. By construction $A={a_1 < a_2 < \\cdots}$\ncontains at least one element from every infinite arithmetic progression, and is a lacunary set, so\nis certainly Sidon.
\n\n AlphaProof has found the following explicit construction: $A = { (n+1)!+n : n\\geq 0}$. This is a\nSidon set, and intersects every arithmetic progression, since for any $a,d\\in \\mathbb{N}$,\n$(a+d+1)!+(a+d)\\in A$, and $d$ divides $(a+d+1)!+d$.
\n\n This was formalized in Lean by Alexeev using Aristotle.
"},"Erdos198.erdos_198.variants.concrete":{"url":"/FormalConjectures/ErdosProblems/«198»/#Erdos198___erdos_198___variants___concrete","anchor":"Erdos198___erdos_198___variants___concrete","docHtml":"\n In fact one such sequence is $n! + n$.
\n\n This was found and proved by AlphaProof.
\n\n It also found $(n + 1)! + n$.
"},"Erdos416.V":{"url":"/FormalConjectures/ErdosProblems/«416»/#Erdos416___V","anchor":"Erdos416___V","docHtml":"\n Let V(x) count the number of n≤x such that ϕ(m)=n is solvable.
\n Let V(x) count the number of n≤x such that ϕ(m)=n is solvable. Does V(2x)/V(x)→2 ?
\n Let V(x) count the number of n≤x such that ϕ(m)=n is solvable.\nIs there an asymptotic formula for V(x)?
\n Let V(x) count the number of n≤x such that ϕ(m)=n is solvable.\nPillai proved V(x)=o(x).\nRef: S. Sivasankaranarayana Pillai,
\n Let V(x) count the number of n≤x such that ϕ(m)=n is solvable.\nErdős proved V(x)=x(logx)^(−1+o(1)).\nRef: Erdős, P.,
\n Let V(x) count the number of n≤x such that ϕ(m)=n is solvable.\nV(x)=x/logx * e^((C+o(1))(log log log x)^2), for some explicit constant C>0.\nRef:Maier, Helmut and Pomerance, Carl,
\n Let V(x) count the number of n≤x such that ϕ(m)=n is solvable.\nV(x) ≍ x/log x*e^(C_1*(log log log x − log log log log x)^2+C_2 log log log x − C_3 log log log log x)\nRef: Ford, Kevin,
\n
\n Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$?
"},"Erdos1059.DecidableIsFactorial":{"url":"/FormalConjectures/ErdosProblems/«1059»/#Erdos1059___DecidableIsFactorial","anchor":"Erdos1059___DecidableIsFactorial"},"Erdos1059.decidableFactorialsLessThanN":{"url":"/FormalConjectures/ErdosProblems/«1059»/#Erdos1059___decidableFactorialsLessThanN","anchor":"Erdos1059___decidableFactorialsLessThanN"},"Erdos1059.DecidableAllFactorialSubtractionsComposite":{"url":"/FormalConjectures/ErdosProblems/«1059»/#Erdos1059___DecidableAllFactorialSubtractionsComposite","anchor":"Erdos1059___DecidableAllFactorialSubtractionsComposite"},"Erdos1059.isFactorial_equivalent":{"url":"/FormalConjectures/ErdosProblems/«1059»/#Erdos1059___isFactorial_equivalent","anchor":"Erdos1059___isFactorial_equivalent"},"Erdos1059.factorialsLessThanN_equivalent":{"url":"/FormalConjectures/ErdosProblems/«1059»/#Erdos1059___factorialsLessThanN_equivalent","anchor":"Erdos1059___factorialsLessThanN_equivalent"},"Erdos1059.allFactorialSubtractionsComposite_equivalent":{"url":"/FormalConjectures/ErdosProblems/«1059»/#Erdos1059___allFactorialSubtractionsComposite_equivalent","anchor":"Erdos1059___allFactorialSubtractionsComposite_equivalent"},"Erdos1059.allFactorialSubtractionsComposite_101":{"url":"/FormalConjectures/ErdosProblems/«1059»/#Erdos1059___allFactorialSubtractionsComposite_101","anchor":"Erdos1059___allFactorialSubtractionsComposite_101"},"Erdos1059.allFactorialSubtractionsComposite_211":{"url":"/FormalConjectures/ErdosProblems/«1059»/#Erdos1059___allFactorialSubtractionsComposite_211","anchor":"Erdos1059___allFactorialSubtractionsComposite_211"},"Erdos1059.notAllFactorialSubtractionsComposite_89":{"url":"/FormalConjectures/ErdosProblems/«1059»/#Erdos1059___notAllFactorialSubtractionsComposite_89","anchor":"Erdos1059___notAllFactorialSubtractionsComposite_89"},"Erdos1059.testFactorialsLessThanN":{"url":"/FormalConjectures/ErdosProblems/«1059»/#Erdos1059___testFactorialsLessThanN","anchor":"Erdos1059___testFactorialsLessThanN"},"Erdos252.erdos_252_sum":{"url":"/FormalConjectures/ErdosProblems/«252»/#Erdos252___erdos_252_sum","anchor":"Erdos252___erdos_252_sum","docHtml":"\n The series ∑ σ k n / n!.
\n Erdős Problem 252: irrationality of the sum for a given $k$.
"},"Erdos252.erdos_252.variants.k_eq_zero":{"url":"/FormalConjectures/ErdosProblems/«252»/#Erdos252___erdos_252___variants___k_eq_zero","anchor":"Erdos252___erdos_252___variants___k_eq_zero","docHtml":"\n∑ σ 0 n / n! is irrational. This is proved in [ErSt71].
\n∑ σ 1 n / n! is irrational. This is proved in [ErSt74].
\n∑ σ 2 n / n! is irrational. This is proved in [ErKa54].
\n∑ σ 3 n / n! is irrational. This is proved in [ScPu06] and [FLC07].
\n∑ σ 4 n / n! is irrational. This is proved in [Pr22].
\n For a fixed k ≥ 5, is ∑ σ k n / n! irrational?.
\n If Schinzel's conjecture is true, then ∑ σ k n / n! is irrational for all k. This is proved\nin [ScPu06].
\n If the prime k-tuples conjecture is true, then ∑ σ k n / n! is irrational. This is proved\nin [FLC07].
\n Let $h(n)$ count the number of powerful integers in $[n^2, (n + 1)^2)$.
"},"Erdos942.erdos_942":{"url":"/FormalConjectures/ErdosProblems/«942»/#Erdos942___erdos_942","anchor":"Erdos942___erdos_942","docHtml":"\n Is there some constant $c > 0$ such that $h(n) < (\\log n)^{c + o(1)}$ and, for infinitely many $n$,\n$h(n) > (\\log n)^{c - o(1)}$.
"},"Erdos942.erdos_942.variants.limsup":{"url":"/FormalConjectures/ErdosProblems/«942»/#Erdos942___erdos_942___variants___limsup","anchor":"Erdos942___erdos_942___variants___limsup","docHtml":"\n It is not hard to prove that $\\limsup h(n) = \\infty$.
"},"Erdos942.erdos_942.variants.density":{"url":"/FormalConjectures/ErdosProblems/«942»/#Erdos942___erdos_942___variants___density","anchor":"Erdos942___erdos_942___variants___density","docHtml":"\n It is not hard to prove that the density $\\delta_l$ of integers for which $h(n) = l$ exists\nand satisfies $$\\sum_l \\delta_l = 1$$.
"},"Erdos624.ExistsEventuallySurjective":{"url":"/FormalConjectures/ErdosProblems/«624»/#Erdos624___ExistsEventuallySurjective","anchor":"Erdos624___ExistsEventuallySurjective","docHtml":"\n The condition that an integer m ensures the existence of a function f covering Fin n\nfor all large enough subsets Y.\nThe property is invariant under bijection, so we use a representative Fin n for a finite set\nof size n.
\n Let $H(n)$ be the minimum integer $m$ such that there is a function $f: \\mathcal{P}(X) \\to X$\nwhere $X$ is a finite set of size $n$, such that for every subset $Y \\subseteq X$ with $|Y| \\ge m$,\nthe set ${f(A) : A \\subseteq Y}$ covers $X$.
"},"Erdos624.erdos_624":{"url":"/FormalConjectures/ErdosProblems/«624»/#Erdos624___erdos_624","anchor":"Erdos624___erdos_624","docHtml":"\n Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function\n$f:{A : A\\subseteq X}\\to X$ so that for every $Y\\subseteq X$ with $\\lvert Y\\rvert \\geq H(n)$\nwe have $\\left{ f(A) : A\\subseteq Y\\right}=X$.\nProve that $H(n)-\\log_2 n \\to \\infty$.
"},"Erdos253.RepresentsAPs":{"url":"/FormalConjectures/ErdosProblems/«253»/#Erdos253___RepresentsAPs","anchor":"Erdos253___RepresentsAPs","docHtml":"\n The predicate that a : ℕ → ℕ is a strictly monotone sequence such that every infinite\narithmetic progression contains infinitely many integers that are the sum of distinct $a_i$s.
\n Let $a_1 < a_2 < \\dotsc$ be an infinite sequence of positive integers such that\n$\\frac{a_{i+1}}{a_i} \\to 1$. If every arithmetic progression contains infinitely many\nintegers which are the sum of distinct $a_i$ then every sufficiently large integer is\nthe sum of distinct $a_i$.
"},"Erdos263.IsIrrationalitySequence":{"url":"/FormalConjectures/ErdosProblems/«263»/#Erdos263___IsIrrationalitySequence","anchor":"Erdos263___IsIrrationalitySequence","docHtml":"\n We call a sequence $a_n$ of positive integers an
\n Note: This is one of many possible notions of \"irrationality sequences\". See\nFormalConjectures/ErdosProblems/264.lean for another possible definition.
"},"Erdos263.erdos_263.parts.i":{"url":"/FormalConjectures/ErdosProblems/«263»/#Erdos263___erdos_263___parts___i","anchor":"Erdos263___erdos_263___parts___i","docHtml":"\n Is $a_n = 2^{2^n}$ an irrationality sequence in the above sense?
"},"Erdos263.erdos_263.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«263»/#Erdos263___erdos_263___parts___ii","anchor":"Erdos263___erdos_263___parts___ii","docHtml":"\n Must every irrationality sequence $a_n$ in the above sense\nsatisfy $a_n^{1/n} \\to \\infty$ as $n \\to \\infty$?\nAnswer: false.
"},"Erdos263.erdos_263.variants.folklore":{"url":"/FormalConjectures/ErdosProblems/«263»/#Erdos263___erdos_263___variants___folklore","anchor":"Erdos263___erdos_263___variants___folklore","docHtml":"\n A folklore result states that any $a_n$ satisfying $\\lim_{n \\to \\infty} a_n^{\\frac{1}{2^n}} = \\infty$\nhas $\\sum \\frac{1}{a_n}$ converging to an irrational number.
"},"Erdos263.erdos_263.variants.sub_doubly_exponential":{"url":"/FormalConjectures/ErdosProblems/«263»/#Erdos263___erdos_263___variants___sub_doubly_exponential","anchor":"Erdos263___erdos_263___variants___sub_doubly_exponential","docHtml":"\n Kovač and Tao [KoTa24] proved that any strictly increasing sequence $a_n$ such that\n$\\sum \\frac{1}{a_n}$ converges and $\\lim \\frac{a_{n+1}}{a_n^2} = 0$ is not\nan irrationality sequence in the above sense.
\n\n [KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series.\narXiv:2406.17593 (2024).
"},"Erdos263.erdos_263.variants.super_doubly_exponential":{"url":"/FormalConjectures/ErdosProblems/«263»/#Erdos263___erdos_263___variants___super_doubly_exponential","anchor":"Erdos263___erdos_263___variants___super_doubly_exponential","docHtml":"\n On the other hand, if there exists some $\\varepsilon > 0$ such that $a_n$ satisfies\n$\\liminf \\frac{a_{n+1}}{a_n^{2+\\varepsilon}} > 0$, then $a_n$ is an irrationality sequence\nby the above folklore result erdos_263.variants.folklore.
\n Koizumi [Ko25] showed that $a_n = \\lfloor \\alpha^{2^n} \\rfloor$ is an irrationality sequence\nfor all but countably many $\\alpha > 1$.
\n\n [Ko25] Koizumi, J., Irrationality of the reciprocal sum of doubly exponential sequences,\narXiv:2504.05933 (2025).
"},"Erdos274.Group.ExactCovering":{"url":"/FormalConjectures/ErdosProblems/«274»/#Erdos274___Group___ExactCovering","anchor":"Erdos274___Group___ExactCovering","docHtml":"\n An exact covering of a group G is a finite collection of subgroups {H_1, ..., H_k} and\nrepresentative {g_1, ..., g_k} such that the cosets g_iH_i are pairwise disjoint and their\nunion covers G.
\n Note that this differs from Partition (α := Subgroup G) because the covering condition there\ninvokes Subgroup.sup which is subgroup generation and thus stronger than union. This definition\nis easier to use in this contect than the alternative Partition (α := Set G), which lacks\nsubgroup definitions such as Subgroup.index.
\n An exact covering of a group G is a finite collection of subgroups {H_1, ..., H_k} and\nrepresentative {g_1, ..., g_k} such that the cosets g_iH_i are pairwise disjoint and their\nunion covers G.
\n Note that this differs from Partition (α := Subgroup G) because the covering condition there\ninvokes Subgroup.sup which is subgroup generation and thus stronger than union. This definition\nis easier to use in this contect than the alternative Partition (α := Set G), which lacks\nsubgroup definitions such as Subgroup.index.
\n An exact covering of a group G is a finite collection of subgroups {H_1, ..., H_k} and\nrepresentative {g_1, ..., g_k} such that the cosets g_iH_i are pairwise disjoint and their\nunion covers G.
\n Note that this differs from Partition (α := Subgroup G) because the covering condition there\ninvokes Subgroup.sup which is subgroup generation and thus stronger than union. This definition\nis easier to use in this contect than the alternative Partition (α := Set G), which lacks\nsubgroup definitions such as Subgroup.index.
\n An exact covering of a group G is a finite collection of subgroups {H_1, ..., H_k} and\nrepresentative {g_1, ..., g_k} such that the cosets g_iH_i are pairwise disjoint and their\nunion covers G.
\n Note that this differs from Partition (α := Subgroup G) because the covering condition there\ninvokes Subgroup.sup which is subgroup generation and thus stronger than union. This definition\nis easier to use in this contect than the alternative Partition (α := Set G), which lacks\nsubgroup definitions such as Subgroup.index.
\n An exact covering of a group G is a finite collection of subgroups {H_1, ..., H_k} and\nrepresentative {g_1, ..., g_k} such that the cosets g_iH_i are pairwise disjoint and their\nunion covers G.
\n Note that this differs from Partition (α := Subgroup G) because the covering condition there\ninvokes Subgroup.sup which is subgroup generation and thus stronger than union. This definition\nis easier to use in this contect than the alternative Partition (α := Set G), which lacks\nsubgroup definitions such as Subgroup.index.
\n An exact covering of a group G is a finite collection of subgroups {H_1, ..., H_k} and\nrepresentative {g_1, ..., g_k} such that the cosets g_iH_i are pairwise disjoint and their\nunion covers G.
\n Note that this differs from Partition (α := Subgroup G) because the covering condition there\ninvokes Subgroup.sup which is subgroup generation and thus stronger than union. This definition\nis easier to use in this contect than the alternative Partition (α := Set G), which lacks\nsubgroup definitions such as Subgroup.index.
\n If $G$ is a group, can there exist an exact covering of $G$ by more than one coset\nof different sizes? (i.e. each element is contained in exactly one of the cosets.)
\n\n The conjectured answer is no: in every such exact covering, two of the subgroups have\nthe same cardinality.
"},"Erdos274.erdos_274.variants.abelian":{"url":"/FormalConjectures/ErdosProblems/«274»/#Erdos274___erdos_274___variants___abelian","anchor":"Erdos274___erdos_274___variants___abelian","docHtml":"\n If G is a finite abelian group then there cannot exist an exact covering of G by more\nthan one cosets of different sizes? (i.e. each element is contained in exactly one\nof the cosets.)
\n Let $G$ be a group, and let $A = {a_1G_1, \\dots, a_kG_k}$ be a finite system of left cosets of\nsubgroups $G_1, \\dots, G_k$ of $G$.
\n\n Herzog and Schönheim conjectured that if $A$ forms a partition of $G$ with $k > 1$, then the\nindices $[G:G_1], \\dots, [G:G_k]$ cannot be distinct.
"},"Erdos508.UnitDistancePlaneGraph":{"url":"/FormalConjectures/ErdosProblems/«508»/#Erdos508___UnitDistancePlaneGraph","anchor":"Erdos508___UnitDistancePlaneGraph","docHtml":"\n The unit-distance graph in the plane, i.e. the graph whose vertices are points in the plane\nand whose edges connect points that are exactly 1 unit apart.
"},"Erdos508.HadwigerNelsonProblem":{"url":"/FormalConjectures/ErdosProblems/«508»/#Erdos508___HadwigerNelsonProblem","anchor":"Erdos508___HadwigerNelsonProblem","docHtml":"\n The Hadwiger–Nelson problem asks: How many colors are required to color the plane\nsuch that no two points at distance 1 from each other have the same color?
"},"Erdos508.HadwigerNelsonAtLeastFive":{"url":"/FormalConjectures/ErdosProblems/«508»/#Erdos508___HadwigerNelsonAtLeastFive","anchor":"Erdos508___HadwigerNelsonAtLeastFive","docHtml":"\n Aubrey de Grey improved the lower bound for the chromatic number of the plane\nto 5 in 2018 using a graph that has >1000 nodes.
\n\n \"The chromatic number of the plane is at least 5\" Aubrey D. N. J. de Grey, 2018\n(https://doi.org/10.48550/arXiv.1804.02385)
"},"Erdos508.HadwigerNelsonAtLeast4":{"url":"/FormalConjectures/ErdosProblems/«508»/#Erdos508___HadwigerNelsonAtLeast4","anchor":"Erdos508___HadwigerNelsonAtLeast4","docHtml":"\n The \"chromatic number of the plane\" is at least 4. This can be\nproven by considering the Moser-Spindel graph\nor the Golomb graph graph.
"},"Erdos508.HadwigerNelsonAtMostSeven":{"url":"/FormalConjectures/ErdosProblems/«508»/#Erdos508___HadwigerNelsonAtMostSeven","anchor":"Erdos508___HadwigerNelsonAtMostSeven","docHtml":"\n This upper bound for the chromatic number of the plane was\nobserved by John R. Isbell. His approach was dividing the\nplane into hexagons of uniform size and coloring them with a repeating\npattern. A proof can probably be found in:
\n\n Soifer, Alexander (2008), The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, New York: Springer, ISBN 978-0-387-74640-1
\n\n An alternative approach that uses square tiling was highlighted by László Székely.
"},"Erdos508.HadwigerNelsonAtLeastThree":{"url":"/FormalConjectures/ErdosProblems/«508»/#Erdos508___HadwigerNelsonAtLeastThree","anchor":"Erdos508___HadwigerNelsonAtLeastThree","docHtml":"\n The chromatic number of the plane is at least 3.
\n\n This is proven by considering an equilateral triangle in the plane.
"},"Erdos884.sumDivisorInvPairwiseDifference":{"url":"/FormalConjectures/ErdosProblems/«884»/#Erdos884___sumDivisorInvPairwiseDifference","anchor":"Erdos884___sumDivisorInvPairwiseDifference","docHtml":"\n The sum $\\sum_{1 \\le i < j \\le \\tau(n)} \\frac{1}{d_j - d_i}$ over all pairs of\ndivisors $d_i < d_j$ of $n$.
"},"Erdos884.sumDivisorInvConsecutiveDifference":{"url":"/FormalConjectures/ErdosProblems/«884»/#Erdos884___sumDivisorInvConsecutiveDifference","anchor":"Erdos884___sumDivisorInvConsecutiveDifference","docHtml":"\n The sum $\\sum_{1 \\le i < \\tau(n)} \\frac{1}{d_{i + 1} - d_i}$ over consecutive\ndivisors of $n$.
"},"Erdos884.Erdos884Prop":{"url":"/FormalConjectures/ErdosProblems/«884»/#Erdos884___Erdos884Prop","anchor":"Erdos884___Erdos884Prop","docHtml":"\n For a natural number n, let $1 = d_1 < \\dotsc < d_{\\tau(n)} = n$ denote the divisors of $n$\nin increasing order.\nDoes it hold that\n$\\sum_{1 \\le i < j \\le \\tau(n)} \\frac{1}{d_j - d_i} \\ll 1 + \\sum_{1 \\le i < \\tau(n)}\n\\frac{1}{d_{i + 1} - d_i}$\nfor $n \\to \\infty`, i.e.\n$\\sum_{1 \\le i < j \\le \\tau(n)} \\frac{1}{d_j - d_i} \\in O \\left( 1 + \\sum_{1 \\le i < \\tau(n)}\n\\frac{1}{d_{i + 1} - d_i}) \\right)$?
\n\n This conjecture has been disproved:
\n\n In September 2025, Terence Tao gave a conditional erdos_884_false_of_hardy_littlewood for this implication.
\n Daniel Larsen subsequently gave an unconditional disproof.
\n\n For a natural number n, let $1 = d_1 < \\dotsc < d_{\\tau(n)} = n$ denote the divisors of $n$\nin increasing order.\nDoes it hold that\n$\\sum_{1 \\le i < j \\le \\tau(n)} \\frac{1}{d_j - d_i} \\ll 1 + \\sum_{1 \\le i < \\tau(n)}\n\\frac{1}{d_{i + 1} - d_i}$\nfor $n \\to \\infty`, i.e.\n$\\sum_{1 \\le i < j \\le \\tau(n)} \\frac{1}{d_j - d_i} \\in O \\left( 1 + \\sum_{1 \\le i < \\tau(n)}\n\\frac{1}{d_{i + 1} - d_i}) \\right)$?
\n\n This conjecture has been disproved:
\n\n In September 2025, Terence Tao gave a conditional erdos_884_false_of_hardy_littlewood for this implication.
\n Daniel Larsen subsequently gave an\nunconditional disproof.
\n\n
\n In September 2025, Terence Tao gave a conditional
\n A triangle is n-cuttable if it can be decomposed into n congruent triangles.
\n A triangle isn't cuttable into zero triangles.
"},"Erdos633.IsCuttable.sq":{"url":"/FormalConjectures/ErdosProblems/«633»/#Erdos633___IsCuttable___sq","anchor":"Erdos633___IsCuttable___sq","docHtml":"\n Every triangle is cuttable into any non-zero square number of congruent triangles.
"},"Erdos633.IsCuttable.of_isSquare":{"url":"/FormalConjectures/ErdosProblems/«633»/#Erdos633___IsCuttable___of_isSquare","anchor":"Erdos633___IsCuttable___of_isSquare","docHtml":"\n Every triangle is cuttable into any non-zero square number of congruent triangles.
"},"Erdos633.isCuttable_iff_isSquare_of_linearIndependent":{"url":"/FormalConjectures/ErdosProblems/«633»/#Erdos633___isCuttable_iff_isSquare_of_linearIndependent","anchor":"Erdos633___isCuttable_iff_isSquare_of_linearIndependent","docHtml":"\n A triangle whose side lengths and angles are integrally independent is cuttable only into\na non-zero square number of congruent triangles. This is proved in [So09c].
"},"Erdos633.erdos_633":{"url":"/FormalConjectures/ErdosProblems/«633»/#Erdos633___erdos_633","anchor":"Erdos633___erdos_633","docHtml":"\n Which triangles can only be decomposed into a square number of congruent triangles?
"},"Erdos633.IsSimiliCuttable":{"url":"/FormalConjectures/ErdosProblems/«633»/#Erdos633___IsSimiliCuttable","anchor":"Erdos633___IsSimiliCuttable","docHtml":"\n A triangle is n-simili-cuttable if it can be decomposed into n similar triangles.
\n A triangle isn't simili-cuttable into zero triangles.
"},"Erdos633.IsSimiliCuttable.of_ne_zero_two_three_five":{"url":"/FormalConjectures/ErdosProblems/«633»/#Erdos633___IsSimiliCuttable___of_ne_zero_two_three_five","anchor":"Erdos633___IsSimiliCuttable___of_ne_zero_two_three_five","docHtml":"\n Every triangle is simili-cuttable into any number of similar triangles, except 0, 2, 3, 5.\nThis is proved in [So09].
"},"Erdos633.exists_isSimiliCuttable_iff_ne_zero_two_three_five":{"url":"/FormalConjectures/ErdosProblems/«633»/#Erdos633___exists_isSimiliCuttable_iff_ne_zero_two_three_five","anchor":"Erdos633___exists_isSimiliCuttable_iff_ne_zero_two_three_five","docHtml":"\n There exists a triangle which isn't simili-cuttable into 0, 2, 3, 5 parts.\nThis is proved in [So09].
"},"Erdos298.erdos_298":{"url":"/FormalConjectures/ErdosProblems/«298»/#Erdos298___erdos_298","anchor":"Erdos298___erdos_298","docHtml":"\n Does every set $A \\subseteq \\mathbb{N}$ of positive density contain some finite $S \\subset A$ such that\n$\\sum_{n \\in S} \\frac{1}{n} = 1$?
\n\n The answer is yes, proved by Bloom [Bl21].
\n\n This was formalized in Lean 3 by Bloom and Mehta.
"},"Erdos298.erdos_298.variants.upper_density":{"url":"/FormalConjectures/ErdosProblems/«298»/#Erdos298___erdos_298___variants___upper_density","anchor":"Erdos298___erdos_298___variants___upper_density","docHtml":"\n In [Bl21] it is proved under the weaker assumption that A only has positive upper density.
\nDivisorSumSet t is the set of natural numbers n such that t can be represented as\na sum of distinct divisors of n.
\n A weaker version of the problem proved by Erdos:\nThe density dₜ of DivisorSumSet (t : ℕ) is bounded from below by 1 / log (t) ^ c₃ and\nfrom above by 1 / log (t) ^ c₄ for some positive constants c₃ and c₄.
\n The density of the divisor sum set is asymptotically equivalent to $c_1 / \\log(t)^{c_2}$.
"},"Erdos859.erdos_859.variants.trivial_case":{"url":"/FormalConjectures/ErdosProblems/«859»/#Erdos859___erdos_859___variants___trivial_case","anchor":"Erdos859___erdos_859___variants___trivial_case","docHtml":"\n A case where we can easily calculate the density of DivisorSumSet t is that of t=0.
\n An easy sanity check is to prove that for every natural number t the density dₜ is\na positive number.\nHint: investigate some multiplicative structure of DivisorSumSet t.
\nRep A m h means m is a sum of at most h elements of Ax.
\n Integers not representable as a finite sum of elements with at most h terms of A\nwhile avoiding n.
\n An asymptotic additive basis of order h is minimal when one cannot obtain an asymptotic\nadditive basis by removing any element from it.
\n Does there exist a minimal basis $A \\subset \\mathbb{N}$ with positive density\nsuch that, for any $n \\in A$, the (upper) density of integers which\ncannot be represented without using $n$ is positive?
"},"Erdos469.Nat.IsSumDivisors":{"url":"/FormalConjectures/ErdosProblems/«469»/#Erdos469___Nat___IsSumDivisors","anchor":"Erdos469___Nat___IsSumDivisors","docHtml":"\n The proposition that n is a sum of distinct proper divisors.
\n Let $A$ be the set of all $n$ such that $n = d_1 + ⋯ + d_k$ with $d_i$ distinct\nproper divisors of $n$, but this is not true for any $m ∣ n$ with $m < n$. Does:\n$$\n\\sum_{n ∈ A} \\frac 1 n\n$$\nconverge?
"},"Erdos40.Erdos40For":{"url":"/FormalConjectures/ErdosProblems/«40»/#Erdos40___Erdos40For","anchor":"Erdos40___Erdos40For","docHtml":"\n The predicate for a function $g\\colon\\mathbb{N} → \\mathbb{R})$ that\n$$\\lvert A\\cap {1,\\ldots,N}\\rvert \\gg \\frac{N^{1/2}}{g(N)}$$\nimplies $\\limsup 1_A\\ast 1_A(n)=\\infty$.
"},"Erdos40.Erdos40ForSet":{"url":"/FormalConjectures/ErdosProblems/«40»/#Erdos40___Erdos40ForSet","anchor":"Erdos40___Erdos40ForSet","docHtml":"\n Given a set of functions $\\mathbb{N} → \\mathbb{R})$, we assert that for all $g$ in that set,\nif $g(N) → \\infty$ then\n$$\\lvert A\\cap {1,\\ldots,N}\\rvert \\gg \\frac{N^{1/2}}{g(N)}$$\nimplies $\\limsup 1_A\\ast 1_A(n)=\\infty$.
"},"Erdos40.erdos_40":{"url":"/FormalConjectures/ErdosProblems/«40»/#Erdos40___erdos_40","anchor":"Erdos40___erdos_40","docHtml":"\n For what functions $g(N) → \\infty$ is it true that\n$$\\lvert A\\cap {1,\\ldots,N}\\rvert \\gg \\frac{N^{1/2}}{g(N)}$$\nimplies $\\limsup 1_A\\ast 1_A(n)=\\infty$?
"},"Erdos40.erdos_40.variants.implies_erdos_28":{"url":"/FormalConjectures/ErdosProblems/«40»/#Erdos40___erdos_40___variants___implies_erdos_28","anchor":"Erdos40___erdos_40___variants___implies_erdos_28","docHtml":"\n If we don't pose additional conditions on the functions, then this is a stronger form of the\nErdős-Turán conjecture, see Erdõs Problem 28,\n(since establishing this for any function $g(N) → \\infty$ would imply a positive solution to Erdős\nProblem 28).
"},"Erdos272.IsArithInterSet":{"url":"/FormalConjectures/ErdosProblems/«272»/#Erdos272___IsArithInterSet","anchor":"Erdos272___IsArithInterSet","docHtml":"\n Let $N \\in\\mathbb{N}$. We say that ${A_1, ..., A_t}\\subseteq\n\\mathcal{P}({1, \\dots, N})$ is an arithmetic intersection set if\n$A_i \\cap A_j$ is a non-empty arithmetic progression for each $i \\neq j$.
"},"Erdos272.maxArithInterCard":{"url":"/FormalConjectures/ErdosProblems/«272»/#Erdos272___maxArithInterCard","anchor":"Erdos272___maxArithInterCard","docHtml":"\n For each $N > 0$, let $t$ be the largest size of an arithmetic\nintersection set.
"},"Erdos272.erdos_272":{"url":"/FormalConjectures/ErdosProblems/«272»/#Erdos272___erdos_272","anchor":"Erdos272___erdos_272","docHtml":"\n Let $N\\geq 1$. What is the largest $t$ such that there are\n$A_1,\\ldots,A_t\\subseteq {1,\\ldots,N}$ with $A_i\\cap A_j$ a non-empty\narithmetic progression for all $i\\neq j$?
"},"Erdos272.erdos_272.variants.isBigO_sq":{"url":"/FormalConjectures/ErdosProblems/«272»/#Erdos272___erdos_272___variants___isBigO_sq","anchor":"Erdos272___erdos_272___variants___isBigO_sq","docHtml":"\n Simonovits and Sós have shown that $t\\ll N^2$.
"},"Erdos272.erdos_272.variants.szabo":{"url":"/FormalConjectures/ErdosProblems/«272»/#Erdos272___erdos_272___variants___szabo","anchor":"Erdos272___erdos_272___variants___szabo","docHtml":"\n Szabo showed that the maximal $t$ is equal to\n$$\n\\frac{N^2}{2} + O(N^{5/3}\\log^3N).\n$$
"},"Erdos272.erdos_272.variants.szabo_strong":{"url":"/FormalConjectures/ErdosProblems/«272»/#Erdos272___erdos_272___variants___szabo_strong","anchor":"Erdos272___erdos_272___variants___szabo_strong","docHtml":"\n Szabo asks whether the maximal $t$ is given by\n$$\n\\frac{N^2}{2} + O(N)\n$$
"},"Erdos346.erdos_346":{"url":"/FormalConjectures/ErdosProblems/«346»/#Erdos346___erdos_346","anchor":"Erdos346___erdos_346","docHtml":"\n Is it true that for every lacunary, strongly complete sequence A that is not complete whenever\ninfinitely many terms are removed from it, lim A (n + 1) / A n = (1 + √5) / 2?
\n We define a sequence f by the formula f n = n.fib - (- 1) ^ n.
\n The sequence f is lacunary.
\n The sequence f is strongly complete, and this is proved in [Gr64d].
\n The sequence f is not complete whenever infinitely many terms are removed from it, and this\nis proved in [Gr64d].
\n Erdős and Graham [ErGr80] remark that it is easy to see that if A (n + 1) / A n > (1 + √5) / 2\nthen the second property is automatically satisfied.
\n Erdős and Graham [ErGr80] also say that it is not hard to construct very irregular sequences\nsatisfying the aforementioned properties.
"},"Erdos857.m":{"url":"/FormalConjectures/ErdosProblems/«857»/#Erdos857___m","anchor":"Erdos857___m","docHtml":"\nm(n, k): minimal sunflower-forcing family size in the non-uniform [n] model.
\n Estimate m(n,k), or better give an asymptotic formula.
\n Let $A \\subset \\mathbb{N}$ be an additive basis of order 2.
\n\n Must there exist $B = {b_1 < b_2 < \\dots} \\subseteq A$ which is also a basis such that\n$\\lim_{k\\to\\infty} \\frac{b_k}{k^2}$ does not exist?
"},"Erdos326.erdos_326.variants.eq":{"url":"/FormalConjectures/ErdosProblems/«326»/#Erdos326___erdos_326___variants___eq","anchor":"Erdos326___erdos_326___variants___eq","docHtml":"\n Erdős originally asked whether this was true with A = B, but this was disproved by Cassels.
\nChowla's cosine problem
\n\n If $A\\subset \\mathbb{N}$ is a finite set of positive integers of size $N > 0$ then is there some\nabsolute constant $c>0$ and $\\theta$ such that\n$$\\sum_{n\\in A}\\cos(n\\theta) < -cN^{1/2}?$$
"},"Erdos510.erdos_510.variants.ruzsa":{"url":"/FormalConjectures/ErdosProblems/«510»/#Erdos510___erdos_510___variants___ruzsa","anchor":"Erdos510___erdos_510___variants___ruzsa","docHtml":"\n Ruzsa [Ru04] proved an upper bound of $-\\exp(O(\\sqrt{\\log N})$.
"},"Erdos510.erdos_510.variants.bedert":{"url":"/FormalConjectures/ErdosProblems/«510»/#Erdos510___erdos_510___variants___bedert","anchor":"Erdos510___erdos_510___variants___bedert","docHtml":"\n Bedert [Be25c] proved an upper bound of $-c N^{1/7}$.
"},"Erdos189.Erdos189For":{"url":"/FormalConjectures/ErdosProblems/«189»/#Erdos189___Erdos189For","anchor":"Erdos189___Erdos189For","docHtml":"\n Erdős problem 189 asked whether the below holds for all rectangles.
"},"Erdos189.erdos_189":{"url":"/FormalConjectures/ErdosProblems/«189»/#Erdos189___erdos_189","anchor":"Erdos189___erdos_189","docHtml":"\n If $\\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the\nvertices of a rectangle of every area?
\n\n Graham, \"On Partitions of 𝔼ⁿ\", Journal of Combinatorial Theory, Series A 28, 89-91 (1980).\n(See \"Concluding Remarks\" on page 96.)
\n\n Solved (with answer False, as formalised below) in:\nVjekoslav Kovač, \"Coloring and density theorems for configurations of a given volume\", 2023\nhttps://arxiv.org/abs/2309.09973\nIn fact, Kovač's colouring is even Jordan measurable (the topological boundary of each\nmonochromatic region is Lebesgue measurable and has measure zero).
\n This was formalized in Lean by Alexeev and Kovac using Aristotle.
"},"Erdos189.erdos_189.variants.square":{"url":"/FormalConjectures/ErdosProblems/«189»/#Erdos189___erdos_189___variants___square","anchor":"Erdos189___erdos_189___variants___square","docHtml":"\n Graham claims this is \"easy to see\".
"},"Erdos189.erdos_189.variants.parallelogram":{"url":"/FormalConjectures/ErdosProblems/«189»/#Erdos189___erdos_189___variants___parallelogram","anchor":"Erdos189___erdos_189___variants___parallelogram","docHtml":"\n Seems to be open, as of January 2025.
"},"Erdos282.greedyUnitFractionRem":{"url":"/FormalConjectures/ErdosProblems/«282»/#Erdos282___greedyUnitFractionRem","anchor":"Erdos282___greedyUnitFractionRem","docHtml":"\n Let $A\\subseteq \\mathbb{N}$ be an infinite set and consider the following\ngreedy algorithm for a rational $x$: choose the minimal $n\\in A$ such\nthat $n\\geq 1/x$ and repeat with $x$ replaced by $x-\\frac{1}{n}$.
\n\n This process of subtracting unit fractions is modelled in greedyUnitFractionRem.\nAt each step t : ℕ, the function greedyUnitFractionRem A x t returns the remainder\nof x with respect to the first t + 1 unit fractions, with denominators taken from A.\nIf this process ever reaches 0 then it terminates. This corresponds to producing a\nrepresentation of x as the sum of distinct unit fractions with denominators from A,\nhowever this function does not return this representation.
\n Let $A\\subseteq \\mathbb{N}$ be an infinite set and consider the following\ngreedy algorithm for a rational $x\\in (0,1)$: choose the minimal $n\\in A$ such\nthat $n\\geq 1/x$ and repeat with $x$ replaced by $x-\\frac{1}{n}$. If this\nterminates after finitely many steps then this produces a representation of\n$x$ as the sum of distinct unit fractions with denominators from $A$.
\n\n Does this process always terminate if $x$ has odd denominator and $A$ is the\nset of odd numbers?
"},"Erdos282.erdos_282.variants.general":{"url":"/FormalConjectures/ErdosProblems/«282»/#Erdos282___erdos_282___variants___general","anchor":"Erdos282___erdos_282___variants___general","docHtml":"\n More generally, for which pairs $x$ and $A$ does this process terminate?
"},"Erdos282.erdos_282.variants.fibonacci":{"url":"/FormalConjectures/ErdosProblems/«282»/#Erdos282___erdos_282___variants___fibonacci","anchor":"Erdos282___erdos_282___variants___fibonacci","docHtml":"\n In 1202 Fibonacci observed that this process terminates for any $x$ when $A=\\mathbb{N}$.
"},"Erdos282.erdos_282.variants.graham":{"url":"/FormalConjectures/ErdosProblems/«282»/#Erdos282___erdos_282___variants___graham","anchor":"Erdos282___erdos_282___variants___graham","docHtml":"\n Graham has shown that $\\frac{m}{n}$ is the sum of distinct unit fractions\nwith denominators $\\equiv a\\pmod{d}$ if and only if\n$$\\left(\\frac{n}{(n,a,d)},\\frac{d}{(a,d)}\\right)=1.$$\nDoes the greedy algorithm always\nterminate in such cases?
"},"Erdos282.greedyUnitFractionRem_sq_one":{"url":"/FormalConjectures/ErdosProblems/«282»/#Erdos282___greedyUnitFractionRem_sq_one","anchor":"Erdos282___greedyUnitFractionRem_sq_one"},"Erdos282.erdos_282.variants.sq":{"url":"/FormalConjectures/ErdosProblems/«282»/#Erdos282___erdos_282___variants___sq","anchor":"Erdos282___erdos_282___variants___sq","docHtml":"\n Graham has also shown that $x$ is the sum of distinct unit fractions with\nsquare denominators if and only if $x\\in [0,\\pi^2/6-1)\\cup [1,\\pi^2/6)$. Does the\ngreedy algorithm for this always terminate? Erdős and Graham believe not - indeed, perhaps it\nfails to terminate almost always.
"},"Erdos968.u":{"url":"/FormalConjectures/ErdosProblems/«968»/#Erdos968___u","anchor":"Erdos968___u","docHtml":"\nu n is the normalized nth prime, defined as pₙ / (n+1) where pₙ is the nth prime\n(with 0.nth Nat.Prime = 2).
\n This corresponds to the classical sequence (p₁/1, p₂/2, p₃/3, ...) while using Nat.nth Prime's\n0-based indexing; in particular, the denominator is always positive.
\n Does the set {n | u n < u (n+1)} have positive natural density?
\n Erdős and Prachar proved ∑_{pₙ < x} |u (n+1) - u n| ≍ (log x)^2 (see [ErPr61]).
\n We encode ∑_{pₙ < x} as a sum over n < Nat.primeCounting' x (the number of primes < x).
\n Erdős and Prachar proved that the set {n | u n > u (n+1)} has positive natural density\n(see [ErPr61]).
\n Erdős asked whether there are infinitely many solutions to uₙ < uₙ₊₁ < uₙ₊₂.
\n Erdős asked whether there are infinitely many solutions to uₙ > uₙ₊₁ > uₙ₊₂.
\n Helper Property: $n$ is the sum of at most $r+1$ numbers, each of which is $r$-full.
"},"Erdos1107.erdos_1107":{"url":"/FormalConjectures/ErdosProblems/«1107»/#Erdos1107___erdos_1107","anchor":"Erdos1107___erdos_1107","docHtml":"\n Let $r \\ge 2$. Is every large integer the sum of at most $r + 1$ many $r$-powerful numbers?
"},"Erdos1107.erdos_1107.variants.two":{"url":"/FormalConjectures/ErdosProblems/«1107»/#Erdos1107___erdos_1107___variants___two","anchor":"Erdos1107___erdos_1107___variants___two","docHtml":"\n Heath-Brown [He88] proved every large integer the sum of at most three $2$-powerful numbers.
"},"Erdos936.EventuallyNotPowerful":{"url":"/FormalConjectures/ErdosProblems/«936»/#Erdos936___EventuallyNotPowerful","anchor":"Erdos936___EventuallyNotPowerful","docHtml":"\n The predicate that a n is only powerful for finitely many n.
\n Is $2^n + 1$ powerful for finitely many $n$?
"},"Erdos936.erdos_936.variants.two_pow_sub_one":{"url":"/FormalConjectures/ErdosProblems/«936»/#Erdos936___erdos_936___variants___two_pow_sub_one","anchor":"Erdos936___erdos_936___variants___two_pow_sub_one","docHtml":"\n Is $2^n - 1$ powerful for finitely many $n$?
"},"Erdos936.erdos_936.variants.factorial_add_one":{"url":"/FormalConjectures/ErdosProblems/«936»/#Erdos936___erdos_936___variants___factorial_add_one","anchor":"Erdos936___erdos_936___variants___factorial_add_one","docHtml":"\n Is $n! + 1$ powerful for finitely many $n$?
"},"Erdos936.erdos_936.variants.factorial_sub_one":{"url":"/FormalConjectures/ErdosProblems/«936»/#Erdos936___erdos_936___variants___factorial_sub_one","anchor":"Erdos936___erdos_936___variants___factorial_sub_one","docHtml":"\n Is $n! - 1$ powerful for finitely many $n$?
"},"Erdos219.primeArithmeticProgressions":{"url":"/FormalConjectures/ErdosProblems/«219»/#Erdos219___primeArithmeticProgressions","anchor":"Erdos219___primeArithmeticProgressions","docHtml":"\n The set of arithmetic progressions of primes
"},"Erdos219.primeArithmeticProgression_3_5_7":{"url":"/FormalConjectures/ErdosProblems/«219»/#Erdos219___primeArithmeticProgression_3_5_7","anchor":"Erdos219___primeArithmeticProgression_3_5_7"},"Erdos219.not_primeArithmeticProgression_1_2":{"url":"/FormalConjectures/ErdosProblems/«219»/#Erdos219___not_primeArithmeticProgression_1_2","anchor":"Erdos219___not_primeArithmeticProgression_1_2"},"Erdos219.empty_not_primeArithmeticProgression":{"url":"/FormalConjectures/ErdosProblems/«219»/#Erdos219___empty_not_primeArithmeticProgression","anchor":"Erdos219___empty_not_primeArithmeticProgression"},"Erdos219.singleton_mem_primeArithmeticProgressions":{"url":"/FormalConjectures/ErdosProblems/«219»/#Erdos219___singleton_mem_primeArithmeticProgressions","anchor":"Erdos219___singleton_mem_primeArithmeticProgressions"},"Erdos219.pair_mem_primeArithmeticProgressions":{"url":"/FormalConjectures/ErdosProblems/«219»/#Erdos219___pair_mem_primeArithmeticProgressions","anchor":"Erdos219___pair_mem_primeArithmeticProgressions"},"Erdos219.erdos_219":{"url":"/FormalConjectures/ErdosProblems/«219»/#Erdos219___erdos_219","anchor":"Erdos219___erdos_219","docHtml":"\n Are there arbitrarily long arithmetic progressions of primes?\nSolution: yes.\nRef: Green, Ben and Tao, Terence,
\nErdős Problem 596 (Erdős–Hajnal, [Er87]). For which graph pairs $(G_1, G_2)$ is it\ntrue that
\n\n (1) for every $n \\geq 1$ there is a graph $H$ without a $G_1$ such that any\n$n$-colouring of $H$'s edges contains a monochromatic $G_2$, and yet\n(2) for every graph $H$ without a $G_1$ there is an $\\aleph_0$-colouring of $H$'s edges\nwith no monochromatic $G_2$?
\n\n Erdős and Hajnal originally conjectured that no such pair exists; but $(C_4, C_6)$\nwitnesses it (Nešetřil–Rödl + Erdős–Hajnal). The full question is to characterise the\nclass of all such pairs, recorded here as answer(sorry).
\n See Problem 595 for the specific case $(G_1, G_2) = (K_4, K_3)$.
"},"Erdos596.erdos_596.variants.exists_exceptional":{"url":"/FormalConjectures/ErdosProblems/«596»/#Erdos596___erdos_596___variants___exists_exceptional","anchor":"Erdos596___erdos_596___variants___exists_exceptional","docHtml":"\n Erdős–Hajnal exceptional pairs exist — recorded as a known direction of erdos_596.
\n Every $C_4$-free graph is a countable union of trees (Erdős–Hajnal [Er87]); trees are\nacyclic, hence $C_6$-free, giving the countable Ramsey escape for $(C_4, C_6)$.
"},"Erdos596.erdos_596.variants.C4_C6_finite_ramsey":{"url":"/FormalConjectures/ErdosProblems/«596»/#Erdos596___erdos_596___variants___C4_C6_finite_ramsey","anchor":"Erdos596___erdos_596___variants___C4_C6_finite_ramsey","docHtml":"\n Nešetřil–Rödl [NeRo75]: for every $n \\geq 1$ there is a $C_4$-free graph whose edges\ncannot be $n$-coloured without a monochromatic $C_6$.
"},"Erdos596.erdos_596.variants.C4_C6_is_exceptional":{"url":"/FormalConjectures/ErdosProblems/«596»/#Erdos596___erdos_596___variants___C4_C6_is_exceptional","anchor":"Erdos596___erdos_596___variants___C4_C6_is_exceptional","docHtml":"\n The pair $(C_4, C_6)$ is Erdős–Hajnal exceptional; combines C4_C6_finite_ramsey and\nC4_free_countable_escape.
\n The original Erdős–Hajnal conjecture (that no exceptional pair exists) is false —\nwitnessed by $(C_4, C_6)$ via C4_C6_is_exceptional.
\n Whether $(K_4, K_3)$ is Erdős–Hajnal exceptional is precisely the content of\nErdős Problem 595. The finite Ramsey property holds (Folkman 1970, Nešetřil–Rödl\n[NeRo75]); the open part is whether every $K_4$-free graph is a countable union of\ntriangle-free graphs.
"},"Erdos596.erdos_596.variants.K4_K3_finite_ramsey":{"url":"/FormalConjectures/ErdosProblems/«596»/#Erdos596___erdos_596___variants___K4_K3_finite_ramsey","anchor":"Erdos596___erdos_596___variants___K4_K3_finite_ramsey","docHtml":"\n Folkman 1970 / Nešetřil–Rödl [NeRo75]: for every $n \\geq 1$ there is a $K_4$-free\ngraph whose edges cannot be $n$-coloured without a monochromatic triangle.
"},"Erdos596.erdos_596.test.empty_is_free":{"url":"/FormalConjectures/ErdosProblems/«596»/#Erdos596___erdos_596___test___empty_is_free","anchor":"Erdos596___erdos_596___test___empty_is_free","docHtml":"\n The empty graph on Fin 0 is Free of any nontrivial subgraph (vacuous). This is the\nsimplest non-trivial witness to G₁.Free H appearing in HasFiniteRamseyProperty.
\n Let $ϕ(n)$ be the Euler's totient function, then the $n$ satisfies $ϕ(n)>ϕ(n - ϕ(n))$\nhave asymptotic density 1.\nReference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\\polhk akowski-{S}chinzel and {E}rd\\H\nos concerning the arithmetical functions {$\\phi$} and\n{$\\sigma$}. Colloq. Math.
"},"Erdos1064.erdos_1064.variants.k2":{"url":"/FormalConjectures/ErdosProblems/«1064»/#Erdos1064___erdos_1064___variants___k2","anchor":"Erdos1064___erdos_1064___variants___k2","docHtml":"\n Let $ϕ(n)$ be the Euler's totient function, there exist infinitely many $n$\nsuch that $ϕ(n)< ϕ(n - ϕ(n))$\nReference: [GLW01] Grytczuk, A. and Luca, F. and W'ojtowicz, M., A conjecture of {E}rdős concerning inequalities for the\n{E}uler totient function.
"},"Erdos1064.erdos_1064.variants.general_function":{"url":"/FormalConjectures/ErdosProblems/«1064»/#Erdos1064___erdos_1064___variants___general_function","anchor":"Erdos1064___erdos_1064___variants___general_function","docHtml":"\n For any function $f(n)=o(n)$,\nwe have $\\phi(n)>\\phi(n-\\phi(n))+f(n)$ for almost all $n$.\nReference:\n[LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\\polhk akowski-{S}chinzel and {E}rd\\H\nos concerning the arithmetical functions {$\\phi$} and\n{$\\sigma$}. Colloq. Math. (2002), 111--130.
"},"Erdos242.erdos_242":{"url":"/FormalConjectures/ErdosProblems/«242»/#Erdos242___erdos_242","anchor":"Erdos242___erdos_242","docHtml":"\n For every $n>2$ there exist distinct integers $1 ≤ x < y < z$\nsuch that $\\frac 4 n = \\frac 1 x + \\frac 1 y + \\frac 1 z$.
"},"Erdos242.erdos_242.variants.schinzel_generalization":{"url":"/FormalConjectures/ErdosProblems/«242»/#Erdos242___erdos_242___variants___schinzel_generalization","anchor":"Erdos242___erdos_242___variants___schinzel_generalization","docHtml":"\n Schinzel conjectured (see [Si56]) the generalisation that, for any fixed $a$, if $n$ is sufficiently\nlarge in terms of $a$ then there exist distinct integers $1\\leq x < y < z$ such that\n$\\frac{a}{n} = \\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}.$
"},"Erdos952.erdos_952":{"url":"/FormalConjectures/ErdosProblems/«952»/#Erdos952___erdos_952","anchor":"Erdos952___erdos_952","docHtml":"\n Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\\ldots$\nsuch that $\\lvert x_{n+1}-x_n\\rvert \\ll 1$?
"},"Erdos260.erdos_260":{"url":"/FormalConjectures/ErdosProblems/«260»/#Erdos260___erdos_260","anchor":"Erdos260___erdos_260","docHtml":"\n Let $a_1 < a_2 < \\cdots$ be an increasing sequence such that $\\frac{a_n}{n} \\to \\infty$.\nIs the sum $\\sum_{n}^{\\infty} \\frac{a_n}{2^{a_n}}$ irrational?
"},"Erdos1113.HasFinitePrimeCoveringSet":{"url":"/FormalConjectures/ErdosProblems/«1113»/#Erdos1113___HasFinitePrimeCoveringSet","anchor":"Erdos1113___HasFinitePrimeCoveringSet","docHtml":"\n A
\n Sierpiński [Si60] proved that there are infinitely many Sierpiński numbers, using covering\nsystems to construct suitable covering sets for any $k$ satisfying a certain congruence.
"},"Erdos1113.erdos_1113":{"url":"/FormalConjectures/ErdosProblems/«1113»/#Erdos1113___erdos_1113","anchor":"Erdos1113___erdos_1113","docHtml":"\nErdős Problem 1113. Do there exist Sierpiński numbers that possess no finite covering set\nof primes?
\n\n Erdős and Graham [ErGr80] conjectured that the answer is yes. A negative answer would imply\nthat there are infinitely many Fermat primes.
"},"Erdos1113.erdos_1113.variants.filaseta_finch_kozek":{"url":"/FormalConjectures/ErdosProblems/«1113»/#Erdos1113___erdos_1113___variants___filaseta_finch_kozek","anchor":"Erdos1113___erdos_1113___variants___filaseta_finch_kozek","docHtml":"\nFilaseta–Finch–Kozek conjecture (2008). Every Sierpiński number is either a perfect power\nor possesses a finite covering set of primes.
"},"Erdos930.IsPower":{"url":"/FormalConjectures/ErdosProblems/«930»/#Erdos930___IsPower","anchor":"Erdos930___IsPower","docHtml":"\n $n$ is a perfect power if there exist natural numbers $m$ and $l$\nsuch that $1 < l$ and $m^l = n$.
"},"Erdos930.erdos_930":{"url":"/FormalConjectures/ErdosProblems/«930»/#Erdos930___erdos_930","anchor":"Erdos930___erdos_930","docHtml":"\n Is it true that, for every $r$, there is a $k$ such that\nif $I_1,\\ldots,I_r$ are disjoint intervals of consecutive integers,\nall of length at least $k$, then\n$$\n\\prod_{1\\leq i\\leq r}\\prod_{m\\in I_i}m\n$$\nis not a perfect power?
"},"Erdos930.nextPrime":{"url":"/FormalConjectures/ErdosProblems/«930»/#Erdos930___nextPrime","anchor":"Erdos930___nextPrime","docHtml":"\n Returns the least prime satisfying $k \\le p$
"},"Erdos930.erdos_930.variants.consecutive_strong":{"url":"/FormalConjectures/ErdosProblems/«930»/#Erdos930___erdos_930___variants___consecutive_strong","anchor":"Erdos930___erdos_930___variants___consecutive_strong","docHtml":"\n Let $k$, $l$, $n$ be integers such that $k \\ge 3$, $l \\ge 2$ and $n + k \\ge p^{(k)}$,\nwhere $p^{(k)}$ is the least prime satisfying $p^{(k)} \\ge k$.\nThen there is a prime $p \\ge k$ for which $l$ does not divide\nthe multiplicity of the prime factor $p$ in $(n + 1) \\ldots (n + k)$.
\n\n Theorem 2 from [ErSe75].
\n\n [ErSe75] Erdős, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.
"},"Erdos930.erdos_930.variants.consecutive_integers":{"url":"/FormalConjectures/ErdosProblems/«930»/#Erdos930___erdos_930___variants___consecutive_integers","anchor":"Erdos930___erdos_930___variants___consecutive_integers","docHtml":"\n Erdos and Selfridge [ErSe75] proved that the product of\nconsecutive integers is never a power (establishing the case $r=1$).
\n\n Theorem 1 from [ErSe75].
\n\n It is implied from erdos_930.variants.consecutive_strong.
\n [ErSe75] Erdős, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.
"},"Erdos247.erdos_247":{"url":"/FormalConjectures/ErdosProblems/«247»/#Erdos247___erdos_247","anchor":"Erdos247___erdos_247","docHtml":"\n Let $n_1 < n_2 < \\cdots$ be a sequence of integers such that\n$$\n\\limsup \\frac{n_k}{k} = \\infty.\n$$\nIs\n$$\n\\sum_{k=1}^{\\infty} \\frac{1}{2^{n_k}}\n$$\ntranscendental?
"},"Erdos247.erdos_247.variants.strong_condition":{"url":"/FormalConjectures/ErdosProblems/«247»/#Erdos247___erdos_247___variants___strong_condition","anchor":"Erdos247___erdos_247___variants___strong_condition","docHtml":"\n Erdős proved the answer is yes under the stronger condition that\n$\\limsup \\frac{n_k}{k^t} = \\infty$ for all $t\\geq 1$.
\n\n [ErGr80] Erdős, P. and Graham, R.,\n
\n Is there a sequence $1 \\le d_1 < d_2 < \\dots$ with density 1 such that all products\n$\\prod_{u \\le i \\le v} d_i$ are distinct?
"},"Erdos868.ncard_add_repr":{"url":"/FormalConjectures/ErdosProblems/«868»/#Erdos868___ncard_add_repr","anchor":"Erdos868___ncard_add_repr","docHtml":"\n The number of ways in which a natural n can be written as the sum of\no members of the set A.
\n Let $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which\n$n$ can be written as the sum of two elements from $A$. If $f(n) \\to \\infty$ as $n \\to \\infty$, then\nmust $A$ contain a minimal additive basis of order $2$?
\n\n Larsen and Larsen [LaLa26] answered this in the negative.
"},"Erdos868.erdos_868.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«868»/#Erdos868___erdos_868___parts___ii","anchor":"Erdos868___erdos_868___parts___ii","docHtml":"\n Let $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which\n$n$ can be written as the sum of two elements from $A$. If $f(n) > \\epsilon \\log n$ for large $n$\nand an arbitrary fixed $\\epsilon > 0$, then must $A$ contain a minimal additive\nbasis of order $2$?
\n\n Larsen and Larsen [LaLa26] constructed a counterexample with $f(n) > c \\log n$ for all large $n$.
"},"Erdos868.erdos_868.variants.fixed_ε":{"url":"/FormalConjectures/ErdosProblems/«868»/#Erdos868___erdos_868___variants___fixed____","anchor":"Erdos868___erdos_868___variants___fixed____","docHtml":"\n Erdős and Nathanson proved that this is true if $f(n) > (\\log \\frac{4}{3})^{-1} \\log n$ for\nall large $n$.
"},"Erdos868.erdos_868.variants.Hartter_Nathanson":{"url":"/FormalConjectures/ErdosProblems/«868»/#Erdos868___erdos_868___variants___Hartter_Nathanson","anchor":"Erdos868___erdos_868___variants___Hartter_Nathanson","docHtml":"\n Härtter and Nathanson proved that there exist additive bases which do not contain\nany minimal additive bases.
"},"Erdos912.h":{"url":"/FormalConjectures/ErdosProblems/«912»/#Erdos912___h","anchor":"Erdos912___h","docHtml":"\n If $n! = \\prod_{i}p_i^{k_i}$ is the factorization into distinct primes, then we define $h(n)$\nto be the number of distinct exponents $k_i$.
"},"Erdos912.erdos_912.variants.selfridge":{"url":"/FormalConjectures/ErdosProblems/«912»/#Erdos912___erdos_912___variants___selfridge","anchor":"Erdos912___erdos_912___variants___selfridge","docHtml":"\n Erdős and Selfridge prove in [Er82c] that $h(n) \\asymp \\left(\\frac{n}{\\log n}\\right)^{1/2}$.
"},"Erdos912.erdos_912":{"url":"/FormalConjectures/ErdosProblems/«912»/#Erdos912___erdos_912","anchor":"Erdos912___erdos_912","docHtml":"\n Prove that there exists some $c>0$ such that\n$$h(n) \\sim c \\left(\\frac{n}{\\log n}\\right)^{1/2}$$\nas $n\\to \\infty$.
"},"Erdos912.erdos_912.variants.tao":{"url":"/FormalConjectures/ErdosProblems/«912»/#Erdos912___erdos_912___variants___tao","anchor":"Erdos912___erdos_912___variants___tao","docHtml":"\n A heuristic of Tao using the Cramér model for the primes suggests this is true with\n$c=\\sqrt{2\\pi}$.
"},"Erdos394.t":{"url":"/FormalConjectures/ErdosProblems/«394»/#Erdos394___t","anchor":"Erdos394___t","docHtml":"\n Let $t_k(n)$ denote the least $m$ such that $n\\mid m(m+1)(m+2)\\cdots (m+k-1).$
"},"Erdos394.erdos_394.parts.i":{"url":"/FormalConjectures/ErdosProblems/«394»/#Erdos394___erdos_394___parts___i","anchor":"Erdos394___erdos_394___parts___i","docHtml":"\n Is it true that $\\sum_{n\\leq x}t_2(n)\\ll \\frac{x^2}{(\\log x)^c}$ for some $c>0$?
"},"Erdos394.erdos_394.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«394»/#Erdos394___erdos_394___parts___ii","anchor":"Erdos394___erdos_394___parts___ii","docHtml":"\n Is it true that, for $k\\geq 2$, $\\sum_{n\\leq x}t_{k+1}(n) =o\\left(\\sum_{n\\leq x}t_k(n)\\right)?$
"},"Erdos394.erdos_394.variants.hall_bound":{"url":"/FormalConjectures/ErdosProblems/«394»/#Erdos394___erdos_394___variants___hall_bound","anchor":"Erdos394___erdos_394___variants___hall_bound","docHtml":"\n In [ErGr80] they mention a conjecture of Erdős that the sum is $o(x^2)$. This was proved by Erdős\nand Hall [ErHa78], who proved that in fact\n$\\sum_{n\\leq x}t_2(n)\\ll \\frac{\\log\\log\\log x}{\\log\\log x}x^2.$
"},"Erdos394.erdos_394.variants.hall_conjecture":{"url":"/FormalConjectures/ErdosProblems/«394»/#Erdos394___erdos_394___variants___hall_conjecture","anchor":"Erdos394___erdos_394___variants___hall_conjecture","docHtml":"\n Erdős and Hall conjecture that the sum is $o(x^2/(\\log x)^c)$ for any $c<\\log 2$.
"},"Erdos394.erdos_394.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«394»/#Erdos394___erdos_394___variants___lower_bound","anchor":"Erdos394___erdos_394___variants___lower_bound","docHtml":"\n Since $t_2(p)=p-1$ for prime $p$ it is trivial that $\\sum_{n\\leq x}t_2(n)\\gg \\frac{x^2}{\\log x}$.
"},"Erdos394.erdos_394.variants.factorial_gap_conjecture":{"url":"/FormalConjectures/ErdosProblems/«394»/#Erdos394___erdos_394___variants___factorial_gap_conjecture","anchor":"Erdos394___erdos_394___variants___factorial_gap_conjecture","docHtml":"\n They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$,\n$t_k(n!)< t_{k-1}(n!)-1$ for all $1\\leq k < n$.
"},"Erdos394.erdos_394.variants.factorial_gap_10":{"url":"/FormalConjectures/ErdosProblems/«394»/#Erdos394___erdos_394___variants___factorial_gap_10","anchor":"Erdos394___erdos_394___variants___factorial_gap_10","docHtml":"\n They proved (with Selfridge) that this holds for $n=10$.
"},"Erdos1135.erdos_1135":{"url":"/FormalConjectures/ErdosProblems/«1135»/#Erdos1135___erdos_1135","anchor":"Erdos1135___erdos_1135","docHtml":"\n The Collatz conjecture states that for any positive integer $n$, there exists a natural\nnumber $m$ such that the $m$-th term of the sequence is 1.
"},"Erdos218.erdos_218.variants.le":{"url":"/FormalConjectures/ErdosProblems/«218»/#Erdos218___erdos_218___variants___le","anchor":"Erdos218___erdos_218___variants___le","docHtml":"\n The set of indices $n$ for which a prime gap is followed by a larger or equal prime gap has a\nnatural density of $\\frac 1 2$.
"},"Erdos218.erdos_218.variants.ge":{"url":"/FormalConjectures/ErdosProblems/«218»/#Erdos218___erdos_218___variants___ge","anchor":"Erdos218___erdos_218___variants___ge","docHtml":"\n The set of indices $n$ for which a prime gap is preceeded by a larger or equal prime gap has a\nnatural density of $\\frac 1 2$.
"},"Erdos218.erdos_218.variants.infinite_equal_prime_gap":{"url":"/FormalConjectures/ErdosProblems/«218»/#Erdos218___erdos_218___variants___infinite_equal_prime_gap","anchor":"Erdos218___erdos_218___variants___infinite_equal_prime_gap","docHtml":"\n There are infintely many indices $n$ such that the prime gap at $n$ is equal to the prime gap\nat $n+1$. This is equivalent to the existence of infinitely many arithmetic progressions of\nlength $3$, see erdos_141.variants.infinite_three.
\n Let $A,B\\subseteq \\mathbb{N}$ such that for all large $N$$$\\lvert A\\cap {1,\\ldots,N}\\rvert \\gg\nN^{1/2}$$and$$\\lvert B\\cap {1,\\ldots,N}\\rvert \\gg N^{1/2}.$$\nIs it true that there are infinitely many solutions to $a_1-a_2=b_1-b_2\\neq 0$ with $a_1,a_2\\in A$\nand $b_1,b_2\\in B$?
\n\n Ruzsa has observed that there is a simple counterexample: take $A$ to be the set of numbers whose\nbinary representation has only non-zero digits in even places, and $B$ similarly but with non-zero\ndigits only in odd places. It is easy to see $A$ and $B$ both grow like $\\gg N^{1/2}$ and yet for\nany $n\\geq 1$ there is exactly one solution to $n=a+b$ with $a\\in A$ and $b\\in B$.
\n\n This was formalized in Lean by van Doorn using Aristotle.
"},"Erdos331.erdos_331.variants.ruzsa":{"url":"/FormalConjectures/ErdosProblems/«331»/#Erdos331___erdos_331___variants___ruzsa","anchor":"Erdos331___erdos_331___variants___ruzsa","docHtml":"\n Ruzsa suggests that a non-trivial variant of this problem arises if one imposes the stronger\ncondition that $|A \\cap {1,\\dots,N}| \\sim c_A N^{1/2}$ for some constant $c_A>0$, and similarly\nfor $B$.
"},"Erdos445.Erdos445Prop":{"url":"/FormalConjectures/ErdosProblems/«445»/#Erdos445___Erdos445Prop","anchor":"Erdos445___Erdos445Prop","docHtml":"\n The property that there exist $a,b\\in(n,n+p^c)$ such that $ab\\equiv 1\\pmod{p}$.
"},"Erdos445.erdos_445":{"url":"/FormalConjectures/ErdosProblems/«445»/#Erdos445___erdos_445","anchor":"Erdos445___erdos_445","docHtml":"\n Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any\n$n\\geq 0$, there exist $a,b\\in(n,n+p^c)$ such that $ab\\equiv 1\\pmod{p}$?
\n\n This is discussed in this MathOverflow question [MathOverflow].
"},"Erdos445.erdos_445.variants.heilbronn":{"url":"/FormalConjectures/ErdosProblems/«445»/#Erdos445___erdos_445___variants___heilbronn","anchor":"Erdos445___erdos_445___variants___heilbronn","docHtml":"\n Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$.
"},"Erdos445.erdos_445.variants.heath_brown":{"url":"/FormalConjectures/ErdosProblems/«445»/#Erdos445___erdos_445___variants___heath_brown","anchor":"Erdos445___erdos_445___variants___heath_brown","docHtml":"\n Heath-Brown [He00] used Kloosterman sums to prove this for all $c>3/4$.
"},"Erdos445.erdos_445.test.small_example":{"url":"/FormalConjectures/ErdosProblems/«445»/#Erdos445___erdos_445___test___small_example","anchor":"Erdos445___erdos_445___test___small_example","docHtml":"\n Small example: for $p=5$, $c=1$, $n=0$, the pair $(2,3) \\in (0,5)$ satisfies\n$2 \\cdot 3 = 6 \\equiv 1 \\pmod{5}$.
"},"Erdos251.erdos_251":{"url":"/FormalConjectures/ErdosProblems/«251»/#Erdos251___erdos_251","anchor":"Erdos251___erdos_251","docHtml":"\n Is $\\sum_{n=1}^\\infty \\frac{p_n}{2^n}$ irrational? Here $p_n$ is the $n$-th prime ($p_1=2, p_2=3, \\dots$).
"},"Erdos853.r":{"url":"/FormalConjectures/ErdosProblems/«853»/#Erdos853___r","anchor":"Erdos853___r","docHtml":"\n
\n Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even\ninteger $t$ such that $d_n = t$ has no solutions for $n \\le x$.
\n\n Is it true that $r(x) \\to \\infty$?
"},"Erdos853.erdos_853.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«853»/#Erdos853___erdos_853___parts___ii","anchor":"Erdos853___erdos_853___parts___ii","docHtml":"\n Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even\ninteger $t$ such that $d_n = t$ has no solutions for $n \\le x$.
\n\n Is it true that $r(x) / \\log x \\to \\infty$?
"},"Erdos68.erdos_68":{"url":"/FormalConjectures/ErdosProblems/«68»/#Erdos68___erdos_68","anchor":"Erdos68___erdos_68","docHtml":"\n Is\n$$\\sum_{n=2}^\\infty \\frac{1}{n!-1}$$\nirrational?
"},"Erdos68.sum_factorial_inv_eq_geometric":{"url":"/FormalConjectures/ErdosProblems/«68»/#Erdos68___sum_factorial_inv_eq_geometric","anchor":"Erdos68___sum_factorial_inv_eq_geometric","docHtml":"\n $$\\sum_{n=2}^\\infty \\frac{1}{n!-1} = \\sum_{n=2}^\\infty \\sum_{k=1}^\\infty \\frac{1}{(n!)^k}$$
"},"Erdos1176.erdos_1176":{"url":"/FormalConjectures/ErdosProblems/«1176»/#Erdos1176___erdos_1176","anchor":"Erdos1176___erdos_1176","docHtml":"\n Let $G$ be a graph with chromatic number $\\aleph_1$. Is it true that there is a colouring of the\nedges with $\\aleph_1$ many colours such that, in any countable colouring of the vertices, there\nexists a vertex colour containing all edge colours?
\n\n A problem of Erdős, Galvin, and Hajnal. The consistency of this was proved by Hajnal and Komjáth.
"},"Erdos590.erdos_590":{"url":"/FormalConjectures/ErdosProblems/«590»/#Erdos590___erdos_590","anchor":"Erdos590___erdos_590","docHtml":"\n Let $α$ be the infinite ordinal $\\omega^{\\omega}$. It was proved by Chang [Ch72] that any red/blue\ncolouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.
"},"Erdos590.erdos_590.variants.two":{"url":"/FormalConjectures/ErdosProblems/«590»/#Erdos590___erdos_590___variants___two","anchor":"Erdos590___erdos_590___variants___two","docHtml":"\n Specker [Sp57] proved that when $α=ω^2$ any red/blue\ncolouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.
"},"Erdos590.erdos_590.variants.ge_three_false":{"url":"/FormalConjectures/ErdosProblems/«590»/#Erdos590___erdos_590___variants___ge_three_false","anchor":"Erdos590___erdos_590___variants___ge_three_false","docHtml":"\n Specker [Sp57] proved that when $α=ω^n$ for $3≤ n < \\omega$ then it is not the case that any\nred/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.
"},"Erdos590.erdos_590.variants.finite_cardinal":{"url":"/FormalConjectures/ErdosProblems/«590»/#Erdos590___erdos_590___variants___finite_cardinal","anchor":"Erdos590___erdos_590___variants___finite_cardinal","docHtml":"\n Let m be a finite cardinal $< \\omega$. Let $α$ be the infinite ordinal $\\omega^{\\omega}$.\nIt was proved by Milnor that any red/blue colouring of the edges of $K_α$ there is either a\nred $K_α$ or a blue $K_3$. A shorter proof was found by Larson [La73]
"},"Erdos61.IsErdosHajnalLowerBound":{"url":"/FormalConjectures/ErdosProblems/«61»/#Erdos61___IsErdosHajnalLowerBound","anchor":"Erdos61___IsErdosHajnalLowerBound","docHtml":"\n
\n The Erdős–Hajnal Conjecture states that there is a constant $c(H) > 0$ for each\n$H$ such that we can take $f(n) = n^{c(H)}$ in the above formulation.
"},"Erdos61.erdos_61.variants.erha89":{"url":"/FormalConjectures/ErdosProblems/«61»/#Erdos61___erdos_61___variants___erha89","anchor":"Erdos61___erdos_61___variants___erha89","docHtml":"\n Erdős and Hajnal [ErHa89] proved that we can take $f(n) = \\exp(c_H \\sqrt{\\log n})$\nfor some constant $c_H > 0$ dependending on $H$.
\n\n [ErHa89] Erdős, P. and Hajnal, A., Ramsey-type theorems. Discrete Appl. Math. (1989), 37-52.
"},"Erdos61.erdos_61.variants.bnss23":{"url":"/FormalConjectures/ErdosProblems/«61»/#Erdos61___erdos_61___variants___bnss23","anchor":"Erdos61___erdos_61___variants___bnss23","docHtml":"\n Bucić, Nguyen, Scott, and Seymour [BNSS23] improved this to\n$f(n) = \\exp(c_H \\sqrt{\\log n \\log \\log n})$ for some constant $c_H > 0$ dependending on $H$.
\n\n [BNSS23] Bucić, M. and Nguyen, T. and Scott, A. and Seymour, P., A loglog step towards Erdos-Hajnal
"},"Erdos413.IsBarrier":{"url":"/FormalConjectures/ErdosProblems/«413»/#Erdos413___IsBarrier","anchor":"Erdos413___IsBarrier","docHtml":"\nIsBarrier f n means n is a barrier for the real-valued function f,\ni.e. (m : ℝ) + f m ≤ (n : ℝ) for all m < n.
\n Are there infinitely many barriers for ω?
\nexpProd n is ∏ kᵢ when n = ∏ pᵢ ^ kᵢ, i.e. the product of the prime exponents of n.
\n Erdős proved that the barrier set for expProd is infinite and even has positive density.
\n Erdős believed there should be infinitely many barriers for Ω, the total prime multiplicity.
\n Selfridge computed that the largest Ω-barrier below 10^5 is 99840.
\n Does there exist some ε > 0 such that there are infinitely many ε-barriers for ω?
\n The set of natural numbers that guarantee a monochromatic arithmetic progression.
\n\n A number N belongs to this set if, for a given number of colors r and an arithmetic\nprogression length k, any r-coloring of the integers {1, ..., N} must contain a\nmonochromatic arithmetic progression of length k.
\n Asserts that for any number of colors r and any progression length k, there\nalways exists some number N large enough to guarantee a monochromatic arithmetic progression.\nIn other words, the set monoAP_guarantee_set is non-empty. This is the fundamental existence\nresult that allows for the definition of the van der Waerden numbers.
\n The van der Waerden number, is the smallest integer N such that any r-coloring of\n{1, ..., N} is guaranteed to contain a monochromatic arithmetic progression of\nlength k. It is defined as the infimum of the (non-empty) set of all such numbers N.
\n An abbreviation for the van der Waerden number for 2 colors, commonly written as W(k).\nThis represents the smallest integer N such that any 2-coloring of {1, ..., N}\nmust contain a monochromatic arithmetic progression of length k.
\n In [Er80] Erdős asks whether\n$$ \\lim_{k \\to \\infty} (W(k))^{1/k} = \\infty $$
"},"Erdos138.erdos_138.variants.prime":{"url":"/FormalConjectures/ErdosProblems/«138»/#Erdos138___erdos_138___variants___prime","anchor":"Erdos138___erdos_138___variants___prime","docHtml":"\n When $p$ is prime Berlekamp [Be68] has proved $W(p+1) ≥ p^{2^p}$.
"},"Erdos138.erdos_138.variants.upper":{"url":"/FormalConjectures/ErdosProblems/«138»/#Erdos138___erdos_138___variants___upper","anchor":"Erdos138___erdos_138___variants___upper","docHtml":"\n Gowers [Go01] has proved $$W(k) \\leq 2^{2^{2^{2^{2^{k+9}}}}.$$
"},"Erdos138.erdos_138.variants.quotient":{"url":"/FormalConjectures/ErdosProblems/«138»/#Erdos138___erdos_138___variants___quotient","anchor":"Erdos138___erdos_138___variants___quotient","docHtml":"\n In [Er81] Erdős asks whether $\\frac{W(k+1)}{W(k)} \\to \\infty$.
"},"Erdos138.erdos_138.variants.difference":{"url":"/FormalConjectures/ErdosProblems/«138»/#Erdos138___erdos_138___variants___difference","anchor":"Erdos138___erdos_138___variants___difference","docHtml":"\n In [Er81] Erdős asks whether $W(k+1) - W(k) \\to \\infty$.
\n\n The DeepMind prover agent has found a formal proof of this statement.
"},"Erdos138.erdos_138.variants.dvd_two_pow":{"url":"/FormalConjectures/ErdosProblems/«138»/#Erdos138___erdos_138___variants___dvd_two_pow","anchor":"Erdos138___erdos_138___variants___dvd_two_pow","docHtml":"\n In [Er80] Erdős asks whether $W(k)/2^k\\to \\infty$.
"},"Erdos1043.levelSet":{"url":"/FormalConjectures/ErdosProblems/«1043»/#Erdos1043___levelSet","anchor":"Erdos1043___levelSet","docHtml":"\n The set ${ z \\in \\mathbb{C} : \\lvert f(z)\\rvert\\leq 1}$
"},"Erdos1043.erdos_1043":{"url":"/FormalConjectures/ErdosProblems/«1043»/#Erdos1043___erdos_1043","anchor":"Erdos1043___erdos_1043","docHtml":"\nErdős Problem 1043:\nLet $f\\in \\mathbb{C}[x]$ be a monic polynomial.\nMust there exist a straight line $\\ell$ such that the projection of\n$${ z: \\lvert f(z)\\rvert\\leq 1}$$\nonto $\\ell$ has measure at most $2$?
\n\n Pommerenke [Po61] proved that the answer is no.
\n\n This was formalized in Lean by Alexeev using Aristotle.
"},"Erdos1043.erdos_1043.variants.weak":{"url":"/FormalConjectures/ErdosProblems/«1043»/#Erdos1043___erdos_1043___variants___weak","anchor":"Erdos1043___erdos_1043___variants___weak","docHtml":"\n On the other hand, Pommerenke also proved there always exists a line such that the projection has\nmeasure at most 3.3.
"},"Erdos701.erdos_701":{"url":"/FormalConjectures/ErdosProblems/«701»/#Erdos701___erdos_701","anchor":"Erdos701___erdos_701","docHtml":"\n Let $\\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if\n$B\\subseteq A\\in\\mathcal{F}$ then $B\\in \\mathcal{F}$). There exists some element $x$ such that\nwhenever $\\mathcal{F}'\\subseteq \\mathcal{F}$ is an intersecting subfamily we have\n$$\\lvert \\mathcal{F}'\\rvert \\leq \\lvert { A\\in \\mathcal{F} : x\\in A}\\rvert.$$
"},"Erdos100.DistancesSeparated":{"url":"/FormalConjectures/ErdosProblems/«100»/#Erdos100___DistancesSeparated","anchor":"Erdos100___DistancesSeparated","docHtml":"\n If two distances in A differ, they differ by at least 1.
"},"Erdos100.erdos_100":{"url":"/FormalConjectures/ErdosProblems/«100»/#Erdos100___erdos_100","anchor":"Erdos100___erdos_100","docHtml":"\n Is the diameter of $A$ at least $Cn$ for some constant $C > 0$?
"},"Erdos100.erdos_100.variants.strong":{"url":"/FormalConjectures/ErdosProblems/«100»/#Erdos100___erdos_100___variants___strong","anchor":"Erdos100___erdos_100___variants___strong","docHtml":"\n Stronger conjecture: diameter $\\geq n - 1$ for sufficiently large $n$.
"},"Erdos100.erdos_100.variants.kanold":{"url":"/FormalConjectures/ErdosProblems/«100»/#Erdos100___erdos_100___variants___kanold","anchor":"Erdos100___erdos_100___variants___kanold","docHtml":"\n From [Kanold]: diameter $\\geq n^{3/4}$.\nTODO: find reference
"},"Erdos100.erdos_100.variants.guth_katz":{"url":"/FormalConjectures/ErdosProblems/«100»/#Erdos100___erdos_100___variants___guth_katz","anchor":"Erdos100___erdos_100___variants___guth_katz","docHtml":"\n From [GuKa15]: diameter $\\gg n / \\log n$.
"},"Erdos100.erdos_100_piepmeyer":{"url":"/FormalConjectures/ErdosProblems/«100»/#Erdos100___erdos_100_piepmeyer","anchor":"Erdos100___erdos_100_piepmeyer","docHtml":"\n From [Piepmeyer]: 9 points with diameter $< 5$.\nTODO: find reference
"},"Erdos971.leastCongruentPrime":{"url":"/FormalConjectures/ErdosProblems/«971»/#Erdos971___leastCongruentPrime","anchor":"Erdos971___leastCongruentPrime","docHtml":"\nleastCongruentPrime a d is the least prime congruent to a modulo d.
\n Let p(a, d) be the least prime congruent to a (mod d).\nDoes there exist a constant c > 0 such that for all large d,\np(a, d) > (1 + c) * φ(d) * log d for ≫ φ(d) many values of a?
\n Erdős [Er49c] proved that the statement in erdos_971 holds for infinitely many values of d.
\n [Er49c] Erdős, P.,
\n Erdős [Er49c] proved that for any ε > 0 we have p(a, d) < ε * φ(d) * log d for ≫_ε φ(d) many\nvalues of a (for all large d).
\n [Er49c] Erdős, P.,
\n The set of lines in $\\mathbb{R}^2$ containing exactly $k$ points from a given set $S$.
"},"Erdos101.numLinesWithFourPointMax":{"url":"/FormalConjectures/ErdosProblems/«101»/#Erdos101___numLinesWithFourPointMax","anchor":"Erdos101___numLinesWithFourPointMax","docHtml":"\n The maximum number of lines containing exactly $4$ points among all sets $S$ of $n$\npoints in $\\mathbb{R}^2$ satisfying the condition that no five points are collinear.
"},"Erdos101.erdos_101":{"url":"/FormalConjectures/ErdosProblems/«101»/#Erdos101___erdos_101","anchor":"Erdos101___erdos_101","docHtml":"\n Given $n$ points in $\\mathbb{R}^2$, no five of which are on a line, the number of\nlines containing four points is $o(n^2)$.
"},"Erdos489.sievedSet":{"url":"/FormalConjectures/ErdosProblems/«489»/#Erdos489___sievedSet","anchor":"Erdos489___sievedSet","docHtml":"\n The set of positive integers not divisible by any element of A.
\n The squared-gap sum ∑_{b_i < x} (b_{i+1} - b_i)², where b_i enumerates the positive\nintegers not divisible by any element of A.
\n Let $A\\subseteq \\mathbb{N}$ be a set such that $\\lvert A\\cap [1,x]\\rvert=o(x^{1/2})$. Let\n$B={ n\\geq 1 : a\\nmid n\\textrm{ for all }a\\in A}$.\nIf $B={b_1 < b_2 < \\cdots}$ then is it true that\n$$\\lim_{x \\to \\infty} \\frac{1}{x}\\sum_{b_i < x}(b_{i+1}-b_i)^2$$\nexists (and is finite)?
\n\n For example, when $A={p^2: p\\textrm{ prime}}$ then $B$ is the set of squarefree numbers,\nand the existence of this limit was proved by Erdős.
\n\n See also [208].
"},"Erdos489.erdos_489.variants.squarefree":{"url":"/FormalConjectures/ErdosProblems/«489»/#Erdos489___erdos_489___variants___squarefree","anchor":"Erdos489___erdos_489___variants___squarefree","docHtml":"\n When $A = {p^2 : p \\textrm{ prime}}$, $B$ is the set of squarefree numbers, and the\nexistence of this limit was proved by Erdős. This is the $\\alpha = 2$ case of Erdős Problem 145.
"},"Erdos822.erdos_822":{"url":"/FormalConjectures/ErdosProblems/«822»/#Erdos822___erdos_822","anchor":"Erdos822___erdos_822","docHtml":"\n Does the set of integers of the form $n + \\varphi(n)$ have positive (lower) density?
\n\n [GIL24] proved this was true.
"},"Erdos275.erdos_275":{"url":"/FormalConjectures/ErdosProblems/«275»/#Erdos275___erdos_275","anchor":"Erdos275___erdos_275","docHtml":"\n If a finite system of $r$ congruences ${ a_i\\pmod{n_i} : 1\\leq i\\leq r}$ (the $n_i$ are not\nnecessarily distinct) covers $2^r$ consecutive integers then it covers all integers.
\n\n This is best possible as the system $2^{i-1}\\pmod{2^i}$ shows. This was proved independently by\nSelfridge and Crittenden and Vanden Eynden [CrVE70].
\n\n This was formalized in Lean by Alexeev using Aristotle.
"},"Set.IsDissociated":{"url":"/FormalConjectures/ErdosProblems/«774»/#Set___IsDissociated","anchor":"Set___IsDissociated","docHtml":"\n We call $A\\subset \\mathbb{N}$ dissociated if $\\sum_{n\\in X}n\\neq \\sum_{m\\in Y}m$\nfor all finite $X,Y\\subset A$ with $X\\neq Y$.
"},"Set.IsProportionatelyDissociated":{"url":"/FormalConjectures/ErdosProblems/«774»/#Set___IsProportionatelyDissociated","anchor":"Set___IsProportionatelyDissociated","docHtml":"\n We call $A$ proportionately dissociated if every finite $B\\subset A$\ncontains a dissociated set of size $\\gg \\lvert B\\rvert$.
\n\n In other words, there is a (global) $c > 0$ such that every finite $B \\subset A$\ncontains a dissociated set of size $\\geq c|B|$.
"},"Erdos774.erdos_774":{"url":"/FormalConjectures/ErdosProblems/«774»/#Erdos774___erdos_774","anchor":"Erdos774___erdos_774","docHtml":"\n Is every proportionately dissociated (infinite) set the union of a finite\nnumber of dissociated sets?
"},"Erdos479.erdos_479":{"url":"/FormalConjectures/ErdosProblems/«479»/#Erdos479___erdos_479","anchor":"Erdos479___erdos_479","docHtml":"\n Is it true that, for all $k\\neq 1$, there are infinitely many $n$ such that\n$2^n\\equiv k\\pmod{n}$?
"},"Erdos427.erdos427":{"url":"/FormalConjectures/ErdosProblems/«427»/#Erdos427___erdos427","anchor":"Erdos427___erdos427","docHtml":"\n The predicate that for every $n$ and $d$, there exists $k$ such that\n$$\nd \\mid p_{n + 1} + \\cdots + p_{n + k},\n$$\nwhere $p_r$ denotes the $r$th prime?
"},"Erdos427.erdos_427":{"url":"/FormalConjectures/ErdosProblems/«427»/#Erdos427___erdos_427","anchor":"Erdos427___erdos_427","docHtml":"\nErdős Problem 427: is it true that, for every $n$ and $d$, there exists $k$ such that\n$$\nd \\mid p_{n + 1} + \\cdots + p_{n + k},\n$$\nwhere $p_r$ denotes the $r$th prime?
"},"Erdos427.ShiuTheorem":{"url":"/FormalConjectures/ErdosProblems/«427»/#Erdos427___ShiuTheorem","anchor":"Erdos427___ShiuTheorem","docHtml":"\n The statement of Shiu's theorem:\nfor any $k \\geq 1$ and $(a, q) = 1$ there exist infinitely many $k$-tuples of consecutive primes\n$p_m, \\dots, p_{m + k - 1}$ all of which are congruent to $a$ modulo $q$.
\n\n [Sh00] Shiu, D. K. L.,
\nShiu's theorem: for any $k \\geq 1$ and $(a, q) = 1$ there exist infinitely many $k$-tuples of consecutive primes\n$p_m, \\dots, p_{m + k - 1}$ all of which are congruent to $a$ modulo $q$.
\n\n [Sh00] Shiu, D. K. L.,
\n Cedric Pilatte has observed that a positive solution to Erdős Problem 427 follows from Shiu's theorem.
"},"Erdos3.erdos_3":{"url":"/FormalConjectures/ErdosProblems/«3»/#Erdos3___erdos_3","anchor":"Erdos3___erdos_3","docHtml":"\n If $A \\subset \\mathbb{N} has $\\sum_{n \\in A}\\frac 1 n = \\infty$, then must $A$ contain arbitrarily\nlong arithmetic progressions?
"},"Erdos316.erdos_316":{"url":"/FormalConjectures/ErdosProblems/«316»/#Erdos316___erdos_316","anchor":"Erdos316___erdos_316","docHtml":"\n Is it true that if $A \\subseteq \\mathbb{N}\\setminus{1}$ is a finite set with\n$\\sum_{n \\in A} \\frac{1}{n} < 2$ then there is a partition $A=A_1 \\sqcup A_2$\nsuch that $\\sum_{n \\in A_i} \\frac{1}{n} < 1$ for $i=1,2$?
\n\n This is not true in general, as shown by Sándor [Sa97].
\n\n The minimal counterexample is ${2,3,4,5,6,7,10,11,13,14,15}$, found by Tom Stobart.
\n\n This was formalized in Lean by Mehta.
"},"Erdos316.erdos_316.variants.multiset":{"url":"/FormalConjectures/ErdosProblems/«316»/#Erdos316___erdos_316___variants___multiset","anchor":"Erdos316___erdos_316___variants___multiset","docHtml":"\n This is not true if $A$ is a multiset, for example $2,3,3,5,5,5,5$.
"},"Erdos316.erdos_316.variants.generalized":{"url":"/FormalConjectures/ErdosProblems/«316»/#Erdos316___erdos_316___variants___generalized","anchor":"Erdos316___erdos_316___variants___generalized","docHtml":"\n This is not true in general, as shown by Sándor [Sa97], who observed that the proper divisors of\n$120$ form a counterexample. More generally, Sándor shows that for any $n\\geq 2$ there exists a\nfinite set $A\\subseteq \\mathbb{N}\\backslash{1}$ with $\\sum_{k\\in A}\\frac{1}{k} < n$ and no\npartition into $n$ parts each of which has $\\sum_{k\\in A_i}\\frac{1}{k}<1$.
"},"Erdos689.erdos_689":{"url":"/FormalConjectures/ErdosProblems/«689»/#Erdos689___erdos_689","anchor":"Erdos689___erdos_689","docHtml":"\n Let n be sufficiently large. Is there some choice of congruence class a_p for all primes\n2 ≤ p ≤ n such that every integer in [1,n] satisfies at least two of the congruences\n≡ a_p (mod p)?
\n Is it true that in any 2-colouring of $\\mathbb{N}$ there exists an infinite set $A$\nsuch that all elements of $A+A$ are the same colour?
\n\n A conjecture of Owings [Ow74].
"},"Erdos1199.erdos_1199.variants.three":{"url":"/FormalConjectures/ErdosProblems/«1199»/#Erdos1199___erdos_1199___variants___three","anchor":"Erdos1199___erdos_1199___variants___three","docHtml":"\n Hindman [Hi79] has shown that this is false for 3-colourings.
"},"Erdos194.erdos_194":{"url":"/FormalConjectures/ErdosProblems/«194»/#Erdos194___erdos_194","anchor":"Erdos194___erdos_194","docHtml":"\n Let $k\\geq 3$. Must any ordering of $\\mathbb{R}$ contain a monotone $k$-term arithmetic progression,\nthat is, some $x_1 <\\cdots < x_k$ which forms an increasing or decreasing $k$-term arithmetic\nprogression?
\n\n Definition of function $f(n) := \\sum_{1\\leq k\\leq n}\\tau(2^k-1)$.\nHere $\\tau$ is the divisor counting function, which is σ 0 in mathlib.
\n Does the limit $\\lim_{n\\to\\infty} \\frac{f(2n)}{f(n)}$ tend to infinity?
\n\n (Other finite limits have been ruled out by [KoLu25], see below)
"},"Erdos893.erdos_893.variants.unbounded":{"url":"/FormalConjectures/ErdosProblems/«893»/#Erdos893___erdos_893___variants___unbounded","anchor":"Erdos893___erdos_893___variants___unbounded","docHtml":"\n Kovač and Luca [KoLu25] (building on a heuristic independently found by\nCambie (personal communication)) have shown that there is no finite limit, in that\n$\\lim_{n\\to\\infty} \\frac{f(2n)}{f(n)}$ is unbounded.
"},"Erdos188.s":{"url":"/FormalConjectures/ErdosProblems/«188»/#Erdos188___s","anchor":"Erdos188___s","docHtml":"\n The set of numbers $k$ such that $\\mathbb{R}^2$ can be red/blue coloured with no pair of red\npoints unit distance apart, and no $k$-term arithmetic progression of blue points with distance 1.
"},"Erdos188.erdos_188":{"url":"/FormalConjectures/ErdosProblems/«188»/#Erdos188___erdos_188","anchor":"Erdos188___erdos_188","docHtml":"\n What is the smallest $k$ such that $\\mathbb{R}^2$ can be red/blue coloured with no pair of red\npoints unit distance apart, and no $k$-term arithmetic progression of blue points with distance 1?
"},"Erdos188.erdos_188.variants.nonempty":{"url":"/FormalConjectures/ErdosProblems/«188»/#Erdos188___erdos_188___variants___nonempty","anchor":"Erdos188___erdos_188___variants___nonempty","docHtml":"\n Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 14, 15):
\n\n It has been shown that there is a large $M$ so that it is possible to partition $\\mathbb{E}^2$ into\ntwo sets $A$ and $B$ so that $A$ contains no pair of points with distance 1 and $B$ contains no A.P.\nof length $M$.
"},"Erdos188.erdos_188.variants.estimate":{"url":"/FormalConjectures/ErdosProblems/«188»/#Erdos188___erdos_188___variants___estimate","anchor":"Erdos188___erdos_188___variants___estimate","docHtml":"\n Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 15):
\n\n How small can $M$ be made? The only estimate currently known is that $M \\le 10000000$ (more or less).\nIn the other direction, it has just been shown by R. Juhász [Ju (79)] that we must have $M \\ge 5$.
"},"Erdos193.IsSWalk":{"url":"/FormalConjectures/ErdosProblems/«193»/#Erdos193___IsSWalk","anchor":"Erdos193___IsSWalk","docHtml":"\n An $S$-walk is a sequence where every difference is in $S$.
"},"Erdos193.HasCollinearTriple":{"url":"/FormalConjectures/ErdosProblems/«193»/#Erdos193___HasCollinearTriple","anchor":"Erdos193___HasCollinearTriple","docHtml":"\n True if set $A$ contains 3 distinct collinear points over $R$.
"},"Erdos193.erdos_193":{"url":"/FormalConjectures/ErdosProblems/«193»/#Erdos193___erdos_193","anchor":"Erdos193___erdos_193","docHtml":"\n Let $S \\subseteq \\mathbb{Z}^3$ be a finite set and let $A = \\lbrace a_1, a_2, \\ldots \\rbrace$ be\nan infinite $S$-walk, so that $a_{i+1} - a_i \\in S$ for all $i$. Must $A$ contain three collinear\npoints?
"},"Erdos193.erdos_193_z2":{"url":"/FormalConjectures/ErdosProblems/«193»/#Erdos193___erdos_193_z2","anchor":"Erdos193___erdos_193_z2","docHtml":"\n [GeRa79] showed that the answer is yes for $\\mathbb{Z}^2$
"},"Erdos985.erdos_985":{"url":"/FormalConjectures/ErdosProblems/«985»/#Erdos985___erdos_985","anchor":"Erdos985___erdos_985","docHtml":"\n Is it true that, for every prime $p$, there is a prime $q \\leq p$ which is a primitive root modulo $p$?
"},"Erdos985.erdos_985.variants.two_three_five_primitive_root":{"url":"/FormalConjectures/ErdosProblems/«985»/#Erdos985___erdos_985___variants___two_three_five_primitive_root","anchor":"Erdos985___erdos_985___variants___two_three_five_primitive_root","docHtml":"\n Heath-Brown proved that at least one of 2, 3, or 5 is a primitive root for infinitely many primes $p$.
"},"Erdos865.erdos_865":{"url":"/FormalConjectures/ErdosProblems/«865»/#Erdos865___erdos_865","anchor":"Erdos865___erdos_865","docHtml":"\n There exists a constant $C>0$ such that, for all large $N$, if $A\\subseteq {1,\\ldots,N}$ has\nsize at least $\\frac{5}{8}N+C$ then there are distinct $a,b,c\\in A$ such that $a+b,a+c,b+c\\in A$.
\n\n A problem of Erdős and Sós (also earlier considered by Choi, Erdős, and Szemerédi [CES75], but Erdős\nhad forgotten this).
"},"Erdos865.erdos_865.variants.k2":{"url":"/FormalConjectures/ErdosProblems/«865»/#Erdos865___erdos_865___variants___k2","anchor":"Erdos865___erdos_865___variants___k2","docHtml":"\n It is a classical folklore fact that if $A\\subseteq {1,\\ldots,2N}$ has size $\\geq N+2$ then\nthere are distinct $a,b\\in A$ such that $a+b\\in A$, which establishes the $k=2$ case.
"},"Erdos865.f":{"url":"/FormalConjectures/ErdosProblems/«865»/#Erdos865___f","anchor":"Erdos865___f"},"Erdos865.erdos_865.variants.sos":{"url":"/FormalConjectures/ErdosProblems/«865»/#Erdos865___erdos_865___variants___sos","anchor":"Erdos865___erdos_865___variants___sos","docHtml":"\n Erdős and Sós conjectured that\n$f_k(N)\\sim \\frac{1}{2}\\left(1+\\sum_{1\\leq r\\leq k-2}\\frac{1}{4^r}\\right) N$,\nwhere $f_k(N)$ is the minimal size of a subset of ${1, \\dots, N}$ guaranteeing $k$ elements\nhave all pairwise sums in the set.
"},"Erdos865.erdos_865.variants.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«865»/#Erdos865___erdos_865___variants___upper_bound","anchor":"Erdos865___erdos_865___variants___upper_bound","docHtml":"\n Choi, Erdős, and Szemerédi [CES75] have proved that, for all $k\\geq 3$, there exists $\\epsilon_k>0$\nsuch that (for large enough $N$) $f_k(N)\\leq \\left(\\frac{2}{3}-\\epsilon_k\\right)N$.
"},"Erdos786.Set.IsMulCardSet":{"url":"/FormalConjectures/ErdosProblems/«786»/#Erdos786___Set___IsMulCardSet","anchor":"Erdos786___Set___IsMulCardSet","docHtml":"\nNat.IsMulCardSet A means that A is a set of natural numbers that\nsatisfies the property that $a_1\\cdots a_r = b_1\\cdots b_s$ with $a_i, b_j\\in A$\ncan only hold when $r = s$.
\n Let $\\epsilon > 0$. Is there some set $A\\subset\\mathbb{N}$ of density $> 1 - \\epsilon$\nsuch that $a_1\\cdots a_r = b_1\\cdots b_s$ with $a_i, b_j\\in A$ can only hold when\n$r = s$?
"},"Erdos786.erdos_786.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«786»/#Erdos786___erdos_786___parts___ii","anchor":"Erdos786___erdos_786___parts___ii","docHtml":"\n Is there some set $A\\subset{1, ..., N}$ of size $\\geq (1 - o(1))N$ such that\n$a_1\\cdots a_r = b_1\\cdots b_s$ with $a_i, b_j\\in A$ can only hold when\n$r = s$?
"},"Erdos786.erdos_786.parts.i.example":{"url":"/FormalConjectures/ErdosProblems/«786»/#Erdos786___erdos_786___parts___i___example","anchor":"Erdos786___erdos_786___parts___i___example","docHtml":"\n An example of such a set with density $\\frac 1 4$ is given by the integers $\\equiv 2\\pmod{4}$
"},"Erdos786.consecutivePrimesFrom":{"url":"/FormalConjectures/ErdosProblems/«786»/#Erdos786___consecutivePrimesFrom","anchor":"Erdos786___consecutivePrimesFrom","docHtml":"\nconsecutivePrimesFrom p k gives the set of k + 1 consecutive primes that are at least p in\nsize. If p is prime then this is the set of k + 1 consecutive primes p, p_1, ..., p_k
\n Let $\\epsilon > 0$ be given. Then, for a sufficiently large prime p, take the sequence of\nconsecutive primes $p_1 < \\cdots < p_k$ such that\n$$\n\\sum_{i=1}^k \\frac{1}{p_i} < 1 < \\sum_{i=1}^{k + 1} \\frac{1}{p_i},\n$$\nand let $A$ be the set of all naturals divisible by exactly one of $p_1, ..., p_k$ (with\nmultiplicity $1$). Then $A$ has density $\\frac{1}{e} - \\epsilon$ and has the property\nthat $a_1\\cdots a_r = b_1\\cdots b_s$ with $a_i, b_j\\in A$ can only hold when $r = s$.
\n Can $\\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone\n3-term arithmetic progressions?
"},"Erdos494.sumMultiset":{"url":"/FormalConjectures/ErdosProblems/«494»/#Erdos494___sumMultiset","anchor":"Erdos494___sumMultiset","docHtml":"\n For a finite set $A \\subset \\mathbb{C}$ and $k \\ge 1$, define $A_k$ as the multiset consisting of\nall sums of $k$ distinct elements of $A$.
"},"Erdos494.Erdos494Unique":{"url":"/FormalConjectures/ErdosProblems/«494»/#Erdos494___Erdos494Unique","anchor":"Erdos494___Erdos494Unique"},"Erdos494.erdos_494.variants.k_eq_2_card_not_pow_two":{"url":"/FormalConjectures/ErdosProblems/«494»/#Erdos494___erdos_494___variants___k_eq_2_card_not_pow_two","anchor":"Erdos494___erdos_494___variants___k_eq_2_card_not_pow_two","docHtml":"\n Selfridge and Straus [SeSt58] showed that the conjecture is true when $k = 2$ and\n$|A| \\ne 2^l$ for $l \\ge 0$.\nThey also gave counterexamples when $k = 2$ and $|A| = 2^l$.
"},"Erdos494.erdos_494.variants.k_eq_2_card_pow_two":{"url":"/FormalConjectures/ErdosProblems/«494»/#Erdos494___erdos_494___variants___k_eq_2_card_pow_two","anchor":"Erdos494___erdos_494___variants___k_eq_2_card_pow_two","docHtml":"\n Selfridge and Straus [SeSt58] gave counterexamples to the conjecture\nwhen $k = 2$ and $|A| = 2^l$.
"},"Erdos494.erdos_494.variants.k_eq_3_card_gt_6":{"url":"/FormalConjectures/ErdosProblems/«494»/#Erdos494___erdos_494___variants___k_eq_3_card_gt_6","anchor":"Erdos494___erdos_494___variants___k_eq_3_card_gt_6","docHtml":"\n Selfridge and Straus [SeSt58] also showed that the conjecture is true when
\n\n $k = 3$ and $|A| > 6$ or
\n\n $k = 4$ and $|A| > 12$.\nMore generally, they proved that $A$ is determined by $A_k$ (and $|A|$) if $|A|$ is divisible by\na prime greater than $k$.
\n\n Selfridge and Straus [SeSt58] showed that the conjecture is true\nwhen $k = 4$ and $|A| > 12$.
"},"Erdos494.erdos_494.variants.card_divisible_by_prime_gt_k":{"url":"/FormalConjectures/ErdosProblems/«494»/#Erdos494___erdos_494___variants___card_divisible_by_prime_gt_k","anchor":"Erdos494___erdos_494___variants___card_divisible_by_prime_gt_k","docHtml":"\n Selfridge and Straus [SeSt58] proved that $A$ is determined by $A_k$\nif $|A|$ is divisible by a prime greater than $k$.
"},"Erdos494.erdos_494.variants.k_eq_card":{"url":"/FormalConjectures/ErdosProblems/«494»/#Erdos494___erdos_494___variants___k_eq_card","anchor":"Erdos494___erdos_494___variants___k_eq_card","docHtml":"\n Kruyt noted that the conjecture fails when $|A| = k$, by rotating $A$ around an appropriate point.
"},"Erdos494.erdos_494.variants.card_eq_2k":{"url":"/FormalConjectures/ErdosProblems/«494»/#Erdos494___erdos_494___variants___card_eq_2k","anchor":"Erdos494___erdos_494___variants___card_eq_2k","docHtml":"\n Similarly, Tao noted that the conjecture fails when $|A| = 2k$, by taking $A$ to be a set of\nthe total sum 0 and considering $-A$.
"},"Erdos494.erdos_494.variants.gordon_fraenkel_straus":{"url":"/FormalConjectures/ErdosProblems/«494»/#Erdos494___erdos_494___variants___gordon_fraenkel_straus","anchor":"Erdos494___erdos_494___variants___gordon_fraenkel_straus","docHtml":"\n Gordon, Fraenkel, and Straus [GRS62] proved that the claim is true for all $k > 2$ when\n$|A|$ is sufficiently large.
"},"Erdos494.prodMultiset":{"url":"/FormalConjectures/ErdosProblems/«494»/#Erdos494___prodMultiset","anchor":"Erdos494___prodMultiset","docHtml":"\n A version in [Er61] by Erdős is product instead of sum, which is false.\nCounterexample (by Steinerberger): consider $k = 3$ and let\n$A = {1, \\zeta_6, \\zeta_6^2, \\zeta_6^4}$ and $B = {1, \\zeta_6^2, \\zeta_6^3, \\zeta_6^4}$.
"},"Erdos494.erdos_494.variants.product":{"url":"/FormalConjectures/ErdosProblems/«494»/#Erdos494___erdos_494___variants___product","anchor":"Erdos494___erdos_494___variants___product","docHtml":"\n A counterexample to the product version of the conjecture (by Steinerberger).
"},"Erdos386.erdos_386":{"url":"/FormalConjectures/ErdosProblems/«386»/#Erdos386___erdos_386","anchor":"Erdos386___erdos_386","docHtml":"\n There is a $k$, such that $2 \\le k \\le n - 2$ and\n$\\binom{n}{k}$ can be the product of consecutive primes infinitely often?
"},"Erdos386.erdos_386.variants.forall":{"url":"/FormalConjectures/ErdosProblems/«386»/#Erdos386___erdos_386___variants___forall","anchor":"Erdos386___erdos_386___variants___forall","docHtml":"\n For all $2 \\le k \\le n - 2$,\ncan $\\binom{n}{k}$ be the product of consecutive primes infinitely often?
"},"Erdos386.erdos_386.variants.two":{"url":"/FormalConjectures/ErdosProblems/«386»/#Erdos386___erdos_386___variants___two","anchor":"Erdos386___erdos_386___variants___two","docHtml":"\n Can $\\binom{n}{2}$ be the product of consecutive primes infinitely often?
"},"Erdos817.g":{"url":"/FormalConjectures/ErdosProblems/«817»/#Erdos817___g","anchor":"Erdos817___g","docHtml":"\n Define $g_k(n)$ to be the minimal $N$ such that ${1, ..., N}$ contains some $A$ of\nsize $|A| = n$ such that\n$$\n\\langle A\\rangle = \\left{\\sum_{a \\in A} \\epsilon_a a : \\epsilon_a \\in{0, 1}\\right}\n$$\ncontains no non-trivial $k$-term arithmetic progression.
"},"Erdos817.erdos_817":{"url":"/FormalConjectures/ErdosProblems/«817»/#Erdos817___erdos_817","anchor":"Erdos817___erdos_817","docHtml":"\n Let $k \\geq 3$. Define $g_k(n)$ to be the minimal $N$ such that\n${1, ..., N}$ contains some $A$ of size $|A| = n$ such that\n$$\n\\langle A\\rangle = \\left{\\sum_{a \\in A} \\epsilon_a a : \\epsilon_a \\in{0, 1}\\right}\n$$\ncontains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In\nparticular, is it true that\n$$\ng_3(n) \\gg 3^n\n$$
"},"Erdos817.erdos_817.variants.bdd_power":{"url":"/FormalConjectures/ErdosProblems/«817»/#Erdos817___erdos_817___variants___bdd_power","anchor":"Erdos817___erdos_817___variants___bdd_power","docHtml":"\n A problem of Erdős and Sárközy who proved\n$$\ng_3(n) \\gg \\frac{3^n}{n^{O(1)}}.\n$$
"},"Erdos750.erdos_750":{"url":"/FormalConjectures/ErdosProblems/«750»/#Erdos750___erdos_750","anchor":"Erdos750___erdos_750","docHtml":"\n Let $f(m)$ be some function such that $f(m)\\to \\infty$ as $m\\to \\infty$. Does there exist a\ngraph $G$ of infinite chromatic number such that every subgraph on $m$ vertices contains\nan independent set of size at least $\\frac{m}{2}-f(m)$?
\n\n Note that in [Er94b] the function $f$ generalises a (proven) result for $f(m) = \\epsilon m$,\nwhere $\\epsilon > 0$. Hence we should assume it is non-negative valued.
"},"Erdos137.erdos_137":{"url":"/FormalConjectures/ErdosProblems/«137»/#Erdos137___erdos_137","anchor":"Erdos137___erdos_137","docHtml":"\n Let $k\\geq 3$. Can the product of any $k$ consecutive integers $N$ ever be powerful? That is,\nmust there always exist a prime $p\\mid N$ such that $p^2\\nmid N$?
"},"Erdos137.erdos_137.variants.perfect_power":{"url":"/FormalConjectures/ErdosProblems/«137»/#Erdos137___erdos_137___variants___perfect_power","anchor":"Erdos137___erdos_137___variants___perfect_power","docHtml":"\n Let $k\\geq 2$. Erdős and Selfridge [ES75] proved that the product of any $k$ consecutive\nintegers $N$ cannot be a perfect power.
\n\n [ES75] P. Erdös, J. L. Selfridge, \"The product of consecutive integers is never a power\",\nIllinois J. Math. 19(2): 292-301, 1975
"},"Erdos137.erdos_137.variants.multiple_powerful_factors":{"url":"/FormalConjectures/ErdosProblems/«137»/#Erdos137___erdos_137___variants___multiple_powerful_factors","anchor":"Erdos137___erdos_137___variants___multiple_powerful_factors","docHtml":"\n Erdős [Er82c] conjectures that, if $k$ is fixed, then for all $n$ sufficiently large and all\npositive integers $m$, there must be at least $k$ distinct primes $p$ such that\n$p\\mid m(m+1)\\cdots (m+n)$ and yet $p^2$ does not divide the right hand side.
\n\n [Er82c] Erdős, Paul, \"Miscellaneous problems in number theory\". Congr. Numer. (1982), 25-45.,
"},"Erdos259.erdos_259":{"url":"/FormalConjectures/ErdosProblems/«259»/#Erdos259___erdos_259","anchor":"Erdos259___erdos_259","docHtml":"\n Is $\\sum_{n} \\mu(n)^2\\frac{n}{2^n}$ irrational?
\n\n This is true, and was proved by Chen and Ruzsa.
\n\n [ChRu99] Chen, Yong-Gao and Ruzsa, Imre Z., On the irrationality of certain series. Period. Math. Hungar. (1999), 31--37.
"},"Erdos740.NoShortOddCycle":{"url":"/FormalConjectures/ErdosProblems/«740»/#Erdos740___NoShortOddCycle","anchor":"Erdos740___NoShortOddCycle","docHtml":"\n A graph avoids odd cycles of length $\\leq r$ if it contains no odd cycles of length at most $r$.
"},"Erdos740.erdos_740":{"url":"/FormalConjectures/ErdosProblems/«740»/#Erdos740___erdos_740","anchor":"Erdos740___erdos_740","docHtml":"\n Let $\\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\\mathfrak{m}$.\nLet $r\\geq 1$. Must $G$ contain a subgraph of chromatic number $\\mathfrak{m}$ which does not contain\nany odd cycle of length $\\leq r$?
"},"_private.0.Erdos340.greedySidon.go":{"url":"/FormalConjectures/ErdosProblems/«340»/#_private___0___Erdos340___greedySidon___go","anchor":"_private___0___Erdos340___greedySidon___go","docHtml":"\n Given a finite Sidon set A and a lower bound m, go finds the smallest number m' ≥ m\nsuch that A ∪ {m'} is Sidon. If A is empty then this returns the value m. Note that\nthe lower bound is required to avoid 0 being a contender in some cases.
\n Main search loop for generating the greedy Sidon sequence. The return value for step n is the\nfinite set of numbers generated so far, a proof that it is Sidon, and the greatest element of\nthe finite set at that point. This is initialised at {1}, then greedySidon.go is\ncalled iteratively using the lower bound max + 1 to find the next smallest Sidon preserving\nnumber.
\ngreedySidon is the sequence obtained by the initial set ${1}$ and iteratively obtaining\nthen next smallest integer that preserves the Sidon property of the set. This gives the\nsequence 1, 2, 4, 8, 13, 21, 31, ....
\n Let $A = {1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \\ldots}$ be the greedy Sidon sequence:\nwe begin with $1$ and iteratively include the next smallest integer that preserves the\nSidon property (i.e. there are no non-trivial solutions to $a + b = c + d$). What is the\norder of growth of $A$? Is it true that $|A \\cap {1, \\ldots, N}| \\gg N^{1/2 - \\varepsilon}$\nfor all $\\varepsilon > 0$ and large $N$?
"},"Erdos340.erdos_340.variants.isTheta":{"url":"/FormalConjectures/ErdosProblems/«340»/#Erdos340___erdos_340___variants___isTheta","anchor":"Erdos340___erdos_340___variants___isTheta","docHtml":"\n Let $A = {1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \\ldots}$ be the greedy Sidon sequence:\nwe begin with $1$ and iteratively include the next smallest integer that preserves the\nSidon property (i.e. there are no non-trivial solutions to $a + b = c + d$). What is the\norder of growth of $A$? Is it true that $|A \\cap {1, \\ldots, N}| \\gg N^{1/2 - \\varepsilon}$\nfor all $\\varepsilon > 0$ and large $N$?
"},"Erdos340.erdos_340.variants.third":{"url":"/FormalConjectures/ErdosProblems/«340»/#Erdos340___erdos_340___variants___third","anchor":"Erdos340___erdos_340___variants___third","docHtml":"\n It is trivial that this sequence grows at least like $\\gg N^{1/3}$.
"},"Erdos340.erdos_340.variants.sub_hasPosDensity":{"url":"/FormalConjectures/ErdosProblems/«340»/#Erdos340___erdos_340___variants___sub_hasPosDensity","anchor":"Erdos340___erdos_340___variants___sub_hasPosDensity","docHtml":"\n Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this has\npositive density.
\n\n [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\ntheory. Monographies de L'Enseignement Mathematique (1980).
"},"Erdos340.erdos_340.variants._22_mem_sub":{"url":"/FormalConjectures/ErdosProblems/«340»/#Erdos340___erdos_340___variants____22_mem_sub","anchor":"Erdos340___erdos_340___variants____22_mem_sub","docHtml":"\n Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this\ncontains $22$, which it does.
\n\n [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\ntheory. Monographies de L'Enseignement Mathematique (1980).
"},"Erdos340.erdos_340.variants._33_mem_sub":{"url":"/FormalConjectures/ErdosProblems/«340»/#Erdos340___erdos_340___variants____33_mem_sub","anchor":"Erdos340___erdos_340___variants____33_mem_sub","docHtml":"\n The smallest integer which is unknown to be in $A - A$ is $33$.
"},"Erdos340.erdos_340.variants.cofinite_sub":{"url":"/FormalConjectures/ErdosProblems/«340»/#Erdos340___erdos_340___variants___cofinite_sub","anchor":"Erdos340___erdos_340___variants___cofinite_sub","docHtml":"\n It may be true that all or almost all integers are in $A - A$.
"},"Erdos340.erdos_340.variants.co_density_zero_sub":{"url":"/FormalConjectures/ErdosProblems/«340»/#Erdos340___erdos_340___variants___co_density_zero_sub","anchor":"Erdos340___erdos_340___variants___co_density_zero_sub","docHtml":"\n It may be true that all or almost all integers are in $A - A$.
"},"Erdos593.erdos_593":{"url":"/FormalConjectures/ErdosProblems/«593»/#Erdos593___erdos_593","anchor":"Erdos593___erdos_593","docHtml":"\nErdős Problem 593 ($500): Characterize those finite 3-uniform hypergraphs which appear\nin every 3-uniform hypergraph of chromatic number $> \\aleph_0$.
\n\n A natural conjectural characterization, recorded here, is that the obligatory finite 3-uniform\nhypergraphs are exactly the 2-colorable ones (Property B). The forward direction\n(IsObligatory → IsTwoColorable) and converse (IsTwoColorable → IsObligatory) are stated as\nseparate variants below; in the graph case ($r = 2$), Erdős–Galvin–Hajnal [EGH75] proved the\nanalogous result (obligatory ⇔ bipartite).
\nErdős Problem 593 — Necessary direction: Every obligatory finite 3-uniform\nhypergraph is 2-colorable.
\n\n This is the natural necessary condition for the conjectural characterization in erdos_593:\nif a finite 3-uniform hypergraph F is not 2-colorable, one expects to construct a\nhypergraph with large chromatic number that contains no copy of F.
\nErdős Problem 593 — Sufficient direction: Every finite 2-colorable 3-uniform\nhypergraph is obligatory.
\n\n This is the converse direction of the erdos_593 characterization: if 2-colorability\nmatches the graph-case characterization (bipartite ⇔ obligatory), then every 2-colorable\nfinite 3-uniform hypergraph must appear in every 3-uniform hypergraph of chromatic number\n$> \\aleph_0$.
\n Together with erdos_593.variants.obligatory_implies_two_colorable, this implies erdos_593.
\nConjunction of the two open implications gives the conjectured characterization: if both\nobligatory_implies_two_colorable and two_colorable_implies_obligatory hold, then the\ncharacterization conjectured in erdos_593 (IsObligatory F ↔ F.IsTwoColorable) follows by\nelementary Iff manipulation.
\nGraph analogue — bipartite graphs are obligatory (Erdős–Galvin–Hajnal [EGH75]):\nFor the 2-uniform (graph) case, a graph of chromatic cardinal $> \\aleph_0$ must contain all\nfinite bipartite graphs. Specifically, for every finite bipartite graph F and every graph\nG with chromatic cardinal $> \\aleph_0$, there is a graph embedding from F into G.
\n This uses Nonempty (F ↪g G) (graph embedding), aligned with the injective vertex map\nused in the hypergraph Appears definition.
\nGraph analogue — no odd cycle is obligatory (Erdős–Galvin–Hajnal [EGH75]):\nFor every odd $k \\geq 3$, there exists a graph with chromatic cardinal $\\aleph_1$ that\ncontains no cycle of length $k$. This shows the class of obligatory graphs is strictly\nsmaller than all finite graphs.
"},"Erdos593.erdos_593.variants.uncountable_vertices_if_large_chromatic":{"url":"/FormalConjectures/ErdosProblems/«593»/#Erdos593___erdos_593___variants___uncountable_vertices_if_large_chromatic","anchor":"Erdos593___erdos_593___variants___uncountable_vertices_if_large_chromatic","docHtml":"\nVertices must be uncountable: Every 3-uniform hypergraph with chromatic cardinal\n$> \\aleph_0$ must have an uncountable vertex set.
\n\nProof: If V is countable, there exists an injection φ : V → ℕ. Using distinct natural\nnumbers as colors gives a proper coloring, so $\\chi(H) \\leq #\\mathbb{N} = \\aleph_0$,\ncontradicting $\\chi(H) > \\aleph_0$.
\nNo hyperedges implies chromatic cardinal ≤ 1: A 3-uniform hypergraph with no edges can\nbe properly colored with a single color, so its chromatic cardinal is at most 1. In\nparticular, $\\chi(H) > \\aleph_0$ implies H has at least one hyperedge.
\nMonotonicity of the obligatory property: If F₁ appears in F₂ and F₂ is obligatory,\nthen F₁ is also obligatory.
\nProof: For any H with $\\chi(H) > \\aleph_0$, since F₂ is obligatory, F₂ appears\nin H via some injection φ₂. Since F₁ appears in F₂ via φ₁, the composition\nφ₂ ∘ φ₁ witnesses that F₁ appears in H.
\nThe empty hypergraph is trivially obligatory: The 3-uniform hypergraph on PEmpty (no\nvertices, no edges) appears in every hypergraph via the empty injection.
\n This degenerate case confirms the definition is well-formed.
"},"Erdos30.h":{"url":"/FormalConjectures/ErdosProblems/«30»/#Erdos30___h","anchor":"Erdos30___h","docHtml":"\n Let $h(N)$ be the maximum size of a Sidon set in ${1, \\dots, N}$.
"},"Erdos30.erdos_30":{"url":"/FormalConjectures/ErdosProblems/«30»/#Erdos30___erdos_30","anchor":"Erdos30___erdos_30","docHtml":"\n Is it true that, for every $\\varepsilon > 0$, $h(N) = \\sqrt N + O_{\\varespilon}(N^\\varespilon)
"},"Erdos324.erdos_324":{"url":"/FormalConjectures/ErdosProblems/«324»/#Erdos324___erdos_324","anchor":"Erdos324___erdos_324","docHtml":"\n Does there exist a polynomial $f(x)\\in\\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with\n$a < b$ nonnegative integers are distinct?
"},"Erdos324.erdos_324.variants.quintic":{"url":"/FormalConjectures/ErdosProblems/«324»/#Erdos324___erdos_324___variants___quintic","anchor":"Erdos324___erdos_324___variants___quintic","docHtml":"\n Probably $f(x) = x^5$ has the property that the sums $f(a)+f(b)$ with\n$a < b$ nonnegative integers are distinct.
"},"Erdos387.erdos_387":{"url":"/FormalConjectures/ErdosProblems/«387»/#Erdos387___erdos_387","anchor":"Erdos387___erdos_387","docHtml":"\n Is there an absolute constant c > 0 such that, for all 1 ≤ k < n, the binomial coefficient\nn.choose k has a divisor in (cn, n]?
\n The following is Schinzel's conjecture, which appears in [Gu04].
"},"Erdos387.erdos_387.variants.easy":{"url":"/FormalConjectures/ErdosProblems/«387»/#Erdos387___erdos_387___variants___easy","anchor":"Erdos387___erdos_387___variants___easy","docHtml":"\n It is easy to see that n.choose k has a divisor in [n / k, n].
\n Is it true for any c < 1 and all n sufficiently large, for all 1 ≤ k < n, n.choose k\nhas a divisor in (cn, n]? This is a variant of erdos_387 and appears in [Gu04].
\n
\n [Er87] Erdős, Paul, Problems and results on set systems and hypergraphs. Extremal problems\nfor finite sets (Visegrád, 1991), Bolyai Soc. Math. Stud. (1994), 217-227.
\n\n [Fo70] Folkman, Jon, Graphs with monochromatic complete subgraphs in every edge coloring.\nSIAM J. Appl. Math. (1970), 19:340-345.
\n\n [NeRo75] Nešetřil, Jaroslav and Rödl, Vojtěch, Type theory of partition problems of graphs.\nRecent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974),\nAcademia, Prague (1975), 405-412.
\n\nErdős Problem 595 ($250): Is there an infinite graph $G$ which contains no $K_4$ and is\nnot the union of countably many triangle-free graphs?
\n\n A problem of Erdős and Hajnal [Er87].
"},"Erdos595.erdos_595.variants.folkman_finite":{"url":"/FormalConjectures/ErdosProblems/«595»/#Erdos595___erdos_595___variants___folkman_finite","anchor":"Erdos595___erdos_595___variants___folkman_finite","docHtml":"\nFolkman–Nešetřil–Rödl (finite version) [Fo70, NeRo75]: For every n ≥ 1, there exists a\ngraph G (on a finite vertex set) that contains no $K_4$ and whose edges cannot be covered by\nn triangle-free graphs.
\n More precisely: for every n : ℕ with 1 ≤ n, there exist a finite type V and a graph\nG : SimpleGraph V with:
\nG.CliqueFree 4 (no $K_4$), and
\n For every family H : Fin n → SimpleGraph V of triangle-free graphs, G ≠ ⨆ i, H i.
\n This is the finite analogue of Problem 595. The proofs of Folkman [Fo70] and Nešetřil–Rödl\n[NeRo75] give different explicit constructions.
"},"Erdos595.erdos_595.variants.subgraph_of_countable_union":{"url":"/FormalConjectures/ErdosProblems/«595»/#Erdos595___erdos_595___variants___subgraph_of_countable_union","anchor":"Erdos595___erdos_595___variants___subgraph_of_countable_union","docHtml":"\nMonotonicity: If G is a countable union of triangle-free graphs and H ≤ G (i.e., H is\na subgraph of G), then H is also a countable union of triangle-free graphs.
\nProof: If G = ⨆ i, G_i with each G_i triangle-free, then H = ⨆ i, H ⊓ G_i.\nEach H ⊓ G_i is triangle-free because it is a subgraph of G_i.
\nTriangle-free graphs are trivially countable unions of triangle-free graphs: if G is\nalready triangle-free, then G = ⨆ i : ℕ, G_i where G_0 = G and G_i = ⊥ for i ≥ 1.
\nThe complete graph ⊤ on ℕ is a countable union of triangle-free graphs: we decompose\nit into the family of star graphs {H_m}_{m : ℕ}, where H_m is the graph with edges {m, n}\nfor all n ≠ m. Each star is triangle-free (any two non-center vertices share no edge within\nthe star), and their union covers all edges of ⊤.
\nProof sketch (star triangle-free): If {a, b, c} were a triangle in H_m, then each of\nthe three edges {a, b}, {a, c}, {b, c} would pass through m. In particular, from\n{a, b} we get a = m or b = m; from {b, c} we get b = m or c = m. Case analysis\nshows that two vertices must equal m, contradicting the triangle having three distinct vertices.
\nThe complete graph ⊤ on Fin 4 is not $K_4$-free: ⊤ on Fin 4 equals the complete\ngraph $K_4$, so it contains $K_4$ as a subgraph and is not $K_4$-free.
\n This sanity check confirms the $K_4$-free hypothesis of Problem 595 is non-trivial.
"},"Erdos595.erdos_595.variants.reformulation_edge_colouring":{"url":"/FormalConjectures/ErdosProblems/«595»/#Erdos595___erdos_595___variants___reformulation_edge_colouring","anchor":"Erdos595___erdos_595___variants___reformulation_edge_colouring","docHtml":"\nReformulation via edge colourings: A graph G is a countable union of triangle-free graphs\nif and only if there is a colouring of the edges of G by ℕ such that no monochromatic\ntriangle exists.
\n More precisely: IsCountableUnionOfTriangleFree G is equivalent to the existence of a map\nc : G.edgeSet → ℕ such that for each n : ℕ, the subgraph of edges coloured n is triangle-free.
\n A set A is \"good\" if it is infinite and there are no distinct a,b,c in A\nsuch that a ∣ (b+c) and b > a, c > a.
\n The set of $p ^ 2$ where $p \\cong 3 \\mod 4$ is prime is an example of a good set.\nFormal proof provided by AlphaProof
"},"Erdos12.erdos_12.parts.i":{"url":"/FormalConjectures/ErdosProblems/«12»/#Erdos12___erdos_12___parts___i","anchor":"Erdos12___erdos_12___parts___i","docHtml":"\n Let $A$ be an infinite set such that there are no distinct $a,b,c \\in A$\nsuch that $a \\mid (b+c)$ and $b,c > a$. Is there such an $A$ with\n$\\liminf \\frac{|A \\cap {1, \\dotsc, N}|}{N^{1/2}} > 0$ ?
\n\n The DeepMind prover agent has found a formal proof of this statement.
"},"Erdos12.erdos_12.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«12»/#Erdos12___erdos_12___parts___ii","anchor":"Erdos12___erdos_12___parts___ii","docHtml":"\n Let $A$ be an infinite set such that there are no distinct $a,b,c \\in A$\nsuch that $a \\mid (b+c)$ and $b,c > a$. Does there exist some absolute constant $c > 0$\nsuch that there are always infinitely many $N$\nwith $|A \\cap {1, \\dotsc, N}| < N^{1−c}$?
\n\n The DeepMind prover agent has found a formal disproof of this statement.
"},"Erdos12.erdos_12.parts.iii":{"url":"/FormalConjectures/ErdosProblems/«12»/#Erdos12___erdos_12___parts___iii","anchor":"Erdos12___erdos_12___parts___iii","docHtml":"\n Let $A$ be an infinite set such that there are no distinct $a,b,c \\in A$\nsuch that $a \\mid (b+c)$ and $b,c > a$. Is it true that $∑_{n \\in A} \\frac{1}{n} < \\infty$?
"},"Erdos12.erdos_12.variants.erdos_sarkozy_density_0":{"url":"/FormalConjectures/ErdosProblems/«12»/#Erdos12___erdos_12___variants___erdos_sarkozy_density_0","anchor":"Erdos12___erdos_12___variants___erdos_sarkozy_density_0","docHtml":"\n Erdős and Sárközy proved that such an $A$ must have density 0.\n[ErSa70] Erd\\H os, P. and Sárk\"ozi, A., On the divisibility properties of sequences of integers.\nProc. London Math. Soc. (3) (1970), 97-101
"},"Erdos12.erdos_12.variants.erdos_sarkozy":{"url":"/FormalConjectures/ErdosProblems/«12»/#Erdos12___erdos_12___variants___erdos_sarkozy","anchor":"Erdos12___erdos_12___variants___erdos_sarkozy","docHtml":"\n Given any function $f(x)\\to \\infty$ as $x\\to \\infty$ there exists a set $A$ with the property\nthat there are no distinct $a,b,c \\in A$ such that $a \\mid (b+c)$ and $b,c > a$, such that there are\ninfinitely many $N$ such that $$\\lvert A\\cap{1,\\ldots,N}\\rvert > \\frac{N}{f(N)}.
"},"Erdos12.erdos_12.variants.example":{"url":"/FormalConjectures/ErdosProblems/«12»/#Erdos12___erdos_12___variants___example","anchor":"Erdos12___erdos_12___variants___example","docHtml":"\n An example of an $A$ with the property that there are no distinct $a,b,c \\in A$ such that\n$a \\mid (b+c)$ and $b,c > a$ and such that\n$$\\liminf \\frac{\\lvert A\\cap{1,\\ldots,N}\\rvert}{N^{1/2}}\\log N > 0$$\nis given by the set of $p^2$, where $p\\equiv 3\\pmod{4}$ is prime.
"},"Erdos12.erdos_12.variants.schoen":{"url":"/FormalConjectures/ErdosProblems/«12»/#Erdos12___erdos_12___variants___schoen","anchor":"Erdos12___erdos_12___variants___schoen","docHtml":"\n Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \\in A$ such\nthat $a \\mid (b+c)$ and $b,c > a$. If all elements in $A$ are pairwise coprime then\n$$\\lvert A\\cap{1,\\ldots,N}\\rvert \\ll N^{2/3}$$
"},"Erdos12.erdos_12.variants.baier":{"url":"/FormalConjectures/ErdosProblems/«12»/#Erdos12___erdos_12___variants___baier","anchor":"Erdos12___erdos_12___variants___baier","docHtml":"\n Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \\in A$ such\nthat $a \\mid (b+c)$ and $b,c > a$. If all elements in $A$ are pairwise coprime then\n$$\\lvert A\\cap{1,\\ldots,N}\\rvert \\ll N^{2/3}/\\log N$$
"},"Erdos539.IsCofactorLowerBound":{"url":"/FormalConjectures/ErdosProblems/«539»/#Erdos539___IsCofactorLowerBound","anchor":"Erdos539___IsCofactorLowerBound","docHtml":"\n We say that $m$ is a cofactor lower bound for a given $n$ if, for every set $A$ of $n$\nnon-negative integers, there are at least $m$ cofactors $a / (a, b)$, where $a, b\\in A$.
"},"Erdos539.cofactorThreshold":{"url":"/FormalConjectures/ErdosProblems/«539»/#Erdos539___cofactorThreshold","anchor":"Erdos539___cofactorThreshold","docHtml":"\n The cofactor threshold $h(n)$, for every positive $n$, is the largest cofactor lower bound\nfor $n$.
"},"Erdos539.erdos_539":{"url":"/FormalConjectures/ErdosProblems/«539»/#Erdos539___erdos_539","anchor":"Erdos539___erdos_539","docHtml":"\n Let $h(n)$ be maximal such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the\nset$$\\left{ \\frac{a}{(a,b)}: a,b\\in A\\right}$$has size at least $h(n)$. Estimate $h(n)$.
"},"Erdos539.erdos_539.variants.sq":{"url":"/FormalConjectures/ErdosProblems/«539»/#Erdos539___erdos_539___variants___sq","anchor":"Erdos539___erdos_539___variants___sq","docHtml":"\n Let $h(n)$ be maximal such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the\nset$$\\left{ \\frac{a}{(a,b)}: a,b\\in A\\right}$$has size at least $h(n)$.\nIs $h(n) = \\Theta(\\sqrt{n})$?
"},"Erdos539.erdos_539.variants.sq_isBigO":{"url":"/FormalConjectures/ErdosProblems/«539»/#Erdos539___erdos_539___variants___sq_isBigO","anchor":"Erdos539___erdos_539___variants___sq_isBigO","docHtml":"\n Erdős and Szemerédi proved that$$n^{1/2} \\ll h(n)$$.
"},"Erdos539.erdos_539.variants.isBigO_sq":{"url":"/FormalConjectures/ErdosProblems/«539»/#Erdos539___erdos_539___variants___isBigO_sq","anchor":"Erdos539___erdos_539___variants___isBigO_sq","docHtml":"\n To prove erdos_539.variants.sq it suffices to show $$ h(n)\\ll n^{1/2}$$.
\n Let $h(n)$ be maximal such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the\nset$$\\left{ \\frac{a}{(a,b)}: a,b\\in A\\right}$$has size at least $h(n)$.\nIs $h(n) = \\Theta(n^{2/3})$?
"},"Erdos539.erdos_539.variants.isBigO_sq_cube_root":{"url":"/FormalConjectures/ErdosProblems/«539»/#Erdos539___erdos_539___variants___isBigO_sq_cube_root","anchor":"Erdos539___erdos_539___variants___isBigO_sq_cube_root","docHtml":"\n Granville and Roesler [GR99] showed that $$h(n)\\ll n^{2/3}$$.
"},"Erdos539.erdos_539.variants.sq_cube_root_isBigO":{"url":"/FormalConjectures/ErdosProblems/«539»/#Erdos539___erdos_539___variants___sq_cube_root_isBigO","anchor":"Erdos539___erdos_539___variants___sq_cube_root_isBigO","docHtml":"\n To prove erdos_539.variants.sq_cube_root it suffices to show $$n^{2/3}\\ll h(n)$$.
\n From [Er73]: The determination of\n$$\n\\lim_{n\\to\\infty}\\frac{\\log(h(n))}{\\log(n)}\n$$\nwill perhaps be not too difficult.
"},"Erdos36.Overlap":{"url":"/FormalConjectures/ErdosProblems/«36»/#Erdos36___Overlap","anchor":"Erdos36___Overlap","docHtml":"\n The number of solutions to the equation $a - b = k$, for $a \\in A$ and $b \\in B$.\nThis represents the \"overlap\" between sets $A$ and $B$ for a given difference $k$.
"},"Erdos36.MaxOverlap":{"url":"/FormalConjectures/ErdosProblems/«36»/#Erdos36___MaxOverlap","anchor":"Erdos36___MaxOverlap","docHtml":"\n The maximum overlap for a given pair of sets $A$ and $B$,\ntaken over all possible integer differences $k$.
"},"Erdos36.M":{"url":"/FormalConjectures/ErdosProblems/«36»/#Erdos36___M","anchor":"Erdos36___M","docHtml":"\n Let $A$ and $B$ be two complementary subsets, a splitting of the numbers ${1, 2, \\dots, 2n}$,\nsuch that both have the same cardinality $n$.\nDefine $M(n)$ to be the minimum MaxOverlap that can be achieved,\nranging over all such partitions $(A, B)$.
\n This example calculates the value of $M 1$. The set is ${1, 2}$, so the only partition is\n$A = {1}, B = {2}$ (or vice versa). The possible differences are $1 - 2 = -1$ and $2 - 1 = 1$.\nThe Overlap for $k=-1$ is 1 (if $A={1}, B={2}$) and for $k=1$ also 1 (if $A={2}, B={1}$ ).\nThe MaxOverlap is $1$, since the Overlap is $0$ for other $k$.\nThus, $M 1 = 1$.
\n The quotient of the minimum maximum overlap $M(N)$ by $N$. The central question of the\nminimum overlap problem is to determine the asymptotic behavior of this quotient as $N \\to \\infty$.
"},"Erdos36.minimum_overlap.variants.lower.erdos_1955":{"url":"/FormalConjectures/ErdosProblems/«36»/#Erdos36___minimum_overlap___variants___lower___erdos_1955","anchor":"Erdos36___minimum_overlap___variants___lower___erdos_1955","docHtml":"\n A lower bound of $\\frac 1 4$.\nSee Some remarks on number theory (in Hebrew)\nby
\n A lower bound of $1 - frac{1}{\\sqrt 2}$.\nScherk (written communication), see\nOn the minimal overlap problem of Erdös\nby
\n A lower bound of $\\frac{4 - \\sqrt{6}}{5}.\nSee On the intersection of a linear set with the translation of its complement\nby
\n A lower bound of $\\sqrt{4 - \\sqrt{15}}$.
"},"Erdos36.minimum_overlap.variants.lower.white_2022":{"url":"/FormalConjectures/ErdosProblems/«36»/#Erdos36___minimum_overlap___variants___lower___white_2022","anchor":"Erdos36___minimum_overlap___variants___lower___white_2022","docHtml":"\n A lower bound of $0.379005$.\nSee Erdős' minimum overlap problem\nby
\n The example (with $N$ even), $A = {\\frac N 2 + 1, \\dots, \\frac{3N}{2}}$\nshows an upper bound of $\\frac 1 2$.
"},"Erdos36.minimum_overlap.variants.upper.MRS_1956":{"url":"/FormalConjectures/ErdosProblems/«36»/#Erdos36___minimum_overlap___variants___upper___MRS_1956","anchor":"Erdos36___minimum_overlap___variants___upper___MRS_1956","docHtml":"\n An upper bound of $\\frac 2 5$.\nSee Minimal overlapping under translation.\nby
\n An upper bound of $0.38200298812318988$.\nSee Advances in the Minimum Overlap Problem\nby
\n An upper bound of $0.3809268534330870$.\nSee The minimum overlap problem\nby
\n Find a better lower bound!
"},"Erdos36.erdos_36.variants.upper":{"url":"/FormalConjectures/ErdosProblems/«36»/#Erdos36___erdos_36___variants___upper","anchor":"Erdos36___erdos_36___variants___upper","docHtml":"\n Find a better upper bound!
"},"Erdos36.erdos_36.variants.exists":{"url":"/FormalConjectures/ErdosProblems/«36»/#Erdos36___erdos_36___variants___exists","anchor":"Erdos36___erdos_36___variants___exists","docHtml":"\n The limit of MinOverlapQuotient exists and it is less than $0.385694$.
\n Find the value of the limit of MinOverlapQuotient!
\n Let f n be the smallest integer for which n! can be represented as the product of distinct\nintegers greater than n, the largest of which is f n.
\nf n - 2 * n = θ (n / log n). This is proved in [EGS82].
\n Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)?
\n Maximum chromatic number of a triangle-free graph on n vertices.
\n Lower bound (Hefty–Horn–King–Pfender 2025).\nThere exists a constant $c_1 \\in (0,1]$ such that, for sufficiently large $n$,\n$$\nc_1 \\sqrt{\\frac{n}{\\log n}} \\le f(n),\n$$\nwhere $f(n)$ denotes the maximum chromatic number of a triangle-free graph on\n$n$ vertices, formalized as triangleFreeMaxChromatic n.
\n Upper bound (Davies–Illingworth 2022).\nThere exists a constant $c_2 \\ge 2$ such that, for sufficiently large $n$,\n$$\nf(n) \\le c_2 \\sqrt{\\frac{n}{\\log n}},\n$$\nwhere $f(n)$ denotes the maximum chromatic number of a triangle-free graph on\n$n$ vertices, formalized as triangleFreeMaxChromatic n.
\n Every triangle-free graph on $5$ vertices can be made bipartite by removing at most $1$ edge.\nThis is the $n = 1$ case of Erdős Problem 23.
"},"Erdos23.erdos_23.variants.n1_tight":{"url":"/FormalConjectures/ErdosProblems/«23»/#Erdos23___erdos_23___variants___n1_tight","anchor":"Erdos23___erdos_23___variants___n1_tight","docHtml":"\n There exists a triangle-free graph on $5$ vertices such that at least $1$ edge must be removed\nto make it bipartite. This shows the bound in erdos_23_n1 is tight.
\n The blow-up of the 5-cycle $C_5$: replace each vertex of $C_5$ with an independent set of $n$\nvertices, and connect two vertices iff their corresponding vertices in $C_5$ are adjacent.\nThe vertex set is $\\mathbb{Z}/5\\mathbb{Z} \\times {0, \\ldots, n-1}$, where $(i, a)$ and $(j, b)$\nare adjacent iff $j = i + 1$ or $i = j + 1$ in $\\mathbb{Z}/5\\mathbb{Z}$.
"},"Erdos23.blowupC5_tight":{"url":"/FormalConjectures/ErdosProblems/«23»/#Erdos23___blowupC5_tight","anchor":"Erdos23___blowupC5_tight","docHtml":"\n The blow-up of $C_5$ shows that the bound $n^2$ in Erdős Problem 23 is tight:\nany bipartite subgraph must omit at least $n^2$ edges.
"},"Erdos23.erdos_23":{"url":"/FormalConjectures/ErdosProblems/«23»/#Erdos23___erdos_23","anchor":"Erdos23___erdos_23","docHtml":"\n Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges?
"},"Erdos364.erdos_364":{"url":"/FormalConjectures/ErdosProblems/«364»/#Erdos364___erdos_364","anchor":"Erdos364___erdos_364","docHtml":"\n There is no consecutive triple of powerful numbers.
"},"Erdos364.erdos_364.variants.strong":{"url":"/FormalConjectures/ErdosProblems/«364»/#Erdos364___erdos_364___variants___strong","anchor":"Erdos364___erdos_364___variants___strong","docHtml":"\n Erdős [Er76d] conjectured a stronger statement: if $n_k$ is the $k$th powerful number,\nthen $n_{k+2} - n_k > n_k^c$ for some constant $c > 0$.
\n\n [Er76d] Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.
"},"Erdos364.erdos_364.variants.weak":{"url":"/FormalConjectures/ErdosProblems/«364»/#Erdos364___erdos_364___variants___weak","anchor":"Erdos364___erdos_364___variants___weak","docHtml":"\n There is no quadruple of powerful numbers, since at least one of the four numbers must be\n$2 \\pmod{4}$, which cannot be powerful (since $2$ divides it, but $2^2$ does not).
"},"Erdos6.erdos_6":{"url":"/FormalConjectures/ErdosProblems/«6»/#Erdos6___erdos_6","anchor":"Erdos6___erdos_6","docHtml":"\n There are infinitely many $n$ such that $d_n < d_{n+1} < d_{n+2}$, where $d$\ndenotes the prime gap function.
"},"Erdos6.erdos_6.variants.increasing":{"url":"/FormalConjectures/ErdosProblems/«6»/#Erdos6___erdos_6___variants___increasing","anchor":"Erdos6___erdos_6___variants___increasing","docHtml":"\n For all $m$, there are infinitely many $n$ such that $d_n < d_{n+1} < \\dots < d_{n+m}$,\nwhere $d$ denotes the prime gap function.
\n\n Proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the\nMaynard-Tao machinery concerning bounded gaps between primes [Ma15]
"},"Erdos6.erdos_6.variants.decreasing":{"url":"/FormalConjectures/ErdosProblems/«6»/#Erdos6___erdos_6___variants___decreasing","anchor":"Erdos6___erdos_6___variants___decreasing","docHtml":"\n For all $m$, there are infinitely many $n$ such that $d_n > d_{n+1} \\dots > d_{n+m}$,\nwhere $d$ denotes the prime gap function.
\n\n Proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the\nMaynard-Tao machinery concerning bounded gaps between primes [Ma15]
"},"Erdos141.Set.IsPrimeProgressionOfLength":{"url":"/FormalConjectures/ErdosProblems/«141»/#Erdos141___Set___IsPrimeProgressionOfLength","anchor":"Erdos141___Set___IsPrimeProgressionOfLength","docHtml":"\n The predicate that a set s consists of l consecutive primes (possibly infinite).\nThis predicate does not assert a specific value for the first term.
\n The first three odd primes are an example of three consecutive primes.
"},"Erdos141.Set.IsAPAndPrimeProgressionOfLength":{"url":"/FormalConjectures/ErdosProblems/«141»/#Erdos141___Set___IsAPAndPrimeProgressionOfLength","anchor":"Erdos141___Set___IsAPAndPrimeProgressionOfLength","docHtml":"\n The predicate that a set s is both an arithmetic progression of length l and a progression\nof l consecutive primes.
\n There are 3 consecutive primes in arithmetic progression.
"},"Erdos141.erdos_141":{"url":"/FormalConjectures/ErdosProblems/«141»/#Erdos141___erdos_141","anchor":"Erdos141___erdos_141","docHtml":"\n Let $k≥3$. Are there $k$ consecutive primes in arithmetic progression?
"},"Erdos141.erdos_141.variants.first_cases":{"url":"/FormalConjectures/ErdosProblems/«141»/#Erdos141___erdos_141___variants___first_cases","anchor":"Erdos141___erdos_141___variants___first_cases","docHtml":"\n The existence of such progressions has been verified for $k≤10$.
"},"Erdos141.erdos_141.variants.eleven":{"url":"/FormalConjectures/ErdosProblems/«141»/#Erdos141___erdos_141___variants___eleven","anchor":"Erdos141___erdos_141___variants___eleven","docHtml":"\n Are there $11$ consecutive primes in arithmetic progression?
"},"Erdos141.consecutivePrimeArithmeticProgressions":{"url":"/FormalConjectures/ErdosProblems/«141»/#Erdos141___consecutivePrimeArithmeticProgressions","anchor":"Erdos141___consecutivePrimeArithmeticProgressions","docHtml":"\n The set of arithmetic progressions of consecutive primes of length $k$.
"},"Erdos141.erdos_141.variants.infinite_three":{"url":"/FormalConjectures/ErdosProblems/«141»/#Erdos141___erdos_141___variants___infinite_three","anchor":"Erdos141___erdos_141___variants___infinite_three","docHtml":"\n It is open, even for $k=3$, whether there are infinitely many such progressions.
"},"Erdos141.erdos_141.variants.infinite_general_case":{"url":"/FormalConjectures/ErdosProblems/«141»/#Erdos141___erdos_141___variants___infinite_general_case","anchor":"Erdos141___erdos_141___variants___infinite_general_case","docHtml":"\n Fix a $k \\geq 3$. Is it true that there are infinitely many arithmetic prime progressions of length $k$?
"},"Erdos707.erdos_707":{"url":"/FormalConjectures/ErdosProblems/«707»/#Erdos707___erdos_707","anchor":"Erdos707___erdos_707","docHtml":"\nErdős Problem 707: It is false that any finite Sidon set can be embedded in a perfect\ndifferent set modulo some $n$.
\n\n As described in [arxiv/2510.19804], a counterexample is provided in [Ha47], see below.\nThe proof of this has been formalized.
\n\n This was formalized in Lean by Alexeev using ChatGPT.
"},"Erdos707.erdos_707.variants.prime_power":{"url":"/FormalConjectures/ErdosProblems/«707»/#Erdos707___erdos_707___variants___prime_power","anchor":"Erdos707___erdos_707___variants___prime_power","docHtml":"\n It is false that any finite Sidon set can be embedded in a perfect\ndifference set modulo p^2 + p + 1 for some prime power p.
\n It is false that any finite Sidon set can be embedded in a perfect\ndifference set modulo p^2 + p + 1 for some prime p.
\n As described in [arxiv/2510.19804], a counterexample is provided in [Ha47], see below.\nThe proof of this has been formalized.
"},"Erdos707.erdos_707.variants.counterexample_prime":{"url":"/FormalConjectures/ErdosProblems/«707»/#Erdos707___erdos_707___variants___counterexample_prime","anchor":"Erdos707___erdos_707___variants___counterexample_prime","docHtml":"\n Alexeev and Mixon [arxiv/2510.19804] have disproved this conjecture, proving that ${1,2,4,8}$\ncannot be extended to a perfect difference set modulo $p^2+p+1$\nfor any prime $p$.
"},"Erdos707.erdos_707.variants.counterexample_mian_chowla":{"url":"/FormalConjectures/ErdosProblems/«707»/#Erdos707___erdos_707___variants___counterexample_mian_chowla","anchor":"Erdos707___erdos_707___variants___counterexample_mian_chowla","docHtml":"\n Alexeev and Mixon [arxiv/2510.19804] have disproved this conjecture,\nshowing that ${1, 2, 4, 8, 13}$ cannot be extended to any perfect difference set.
"},"Erdos707.erdos_707.variants.counterexample_hall":{"url":"/FormalConjectures/ErdosProblems/«707»/#Erdos707___erdos_707___variants___counterexample_hall","anchor":"Erdos707___erdos_707___variants___counterexample_hall","docHtml":"\n This conjecture was actually first disproved by Hall in 1947 [Ha47], long before Erdős asked\nthis question.\nA counterexample for any modulus from from [Ha47] in the paragraph following Theorem 4.3, where it\nwas given as ${-8, -6, 0, 1, 4}$, but this can be shifted to natural numbers\nas pointed out in [arxiv/2510.19804].
"},"Erdos707.erdos_707.variants.perfect_difference_set_size_bound":{"url":"/FormalConjectures/ErdosProblems/«707»/#Erdos707___erdos_707___variants___perfect_difference_set_size_bound","anchor":"Erdos707___erdos_707___variants___perfect_difference_set_size_bound","docHtml":"\n A perfect difference set modulo n must have size ≤ √n + 1.
\n The Singer construction gives perfect difference sets for n = p^2 + p + 1 where p is a\nprime power.
\n The set {1, 2, 4} is a Sidon set.
\n The set {1, 2, 4} can be embedded in a perfect difference set modulo 7.
\n For small Sidon sets, we can check the conjecture directly.
"},"Erdos359.IsGoodFor":{"url":"/FormalConjectures/ErdosProblems/«359»/#Erdos359___IsGoodFor","anchor":"Erdos359___IsGoodFor","docHtml":"\n The predicate that A is monotone, A 0 = n and for all j, A (j + 1) is the smallest natural number that\ncannot be written as a sum of consecutive terms of A 0, ..., A j
\n Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the\nleast integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k \\to \\infty$.
"},"Erdos359.erdos_359.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«359»/#Erdos359___erdos_359___parts___ii","anchor":"Erdos359___erdos_359___parts___ii","docHtml":"\n Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the\nleast integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k ^ {1 + c} \\to 0$\nfor any $c > 0$.
"},"Erdos359.erdos_359.variants.isGoodFor_1_low_values":{"url":"/FormalConjectures/ErdosProblems/«359»/#Erdos359___erdos_359___variants___isGoodFor_1_low_values","anchor":"Erdos359___erdos_359___variants___isGoodFor_1_low_values","docHtml":"\n Suppose monotone sequence $A$ satisfies the following: A 0 = 1 and for all j, A (j + 1) is the\nsmallest natural number that cannot be written as a sum of consecutive terms of A 0, ..., A j.\nThen the first few terms of $A$ are $1,2,4,5,8,10,14,15,...$.
\n Suppose monotone sequence $A$ satisfies the following: A 0 = 1 and for all j, A (j + 1) is the\nsmallest natural number that cannot be written as a sum of consecutive terms of A 0, ..., A j.\nThen it is conjectured that $$a_k ~ \\frac{k \\log k}{\\log \\log k}$$.
\n Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large\nfinite multiset of integers with $\\sum_{n \\in A} 1/n > K$ there exists some $S \\subseteq A$ such that\n$1 - \\exp(-(c*K)) < \\sum_{n \\in S} 1/n \\le 1$?
\n A sequence $a_n$ of integers is called an irrationality sequence if for every bounded sequence of integers $b_n$ with $a_n + b_n \\neq 0$ and\n$b_n \\neq 0$ for all $n$, the sum\n$$\n\\sum \\frac{1}{a_n + b_n}\n$$\nis irrational.
\n\n Note: there are other possible definitions of this concept. See\nFormalConjectures/ErdosProblems/263.lean for another possible definition.
"},"Erdos264.erdos_264.parts.i":{"url":"/FormalConjectures/ErdosProblems/«264»/#Erdos264___erdos_264___parts___i","anchor":"Erdos264___erdos_264___parts___i","docHtml":"\n Is $2^n$ an example of an irrationality sequence? Kovač and Tao proved that it is not [KoTa24]
\n\n [KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).
"},"Erdos264.erdos_264.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«264»/#Erdos264___erdos_264___parts___ii","anchor":"Erdos264___erdos_264___parts___ii","docHtml":"\n Is $n!$ an example of an irrationality sequence?
"},"Erdos264.erdos_264.variants.example":{"url":"/FormalConjectures/ErdosProblems/«264»/#Erdos264___erdos_264___variants___example","anchor":"Erdos264___erdos_264___variants___example","docHtml":"\n One example is $2^{2^n}$.
"},"Erdos264.erdos_264.variants.ko_tao_neg":{"url":"/FormalConjectures/ErdosProblems/«264»/#Erdos264___erdos_264___variants___ko_tao_neg","anchor":"Erdos264___erdos_264___variants___ko_tao_neg","docHtml":"\n Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers\n$a_n$ such that $\\sum \\frac{1}{a_n}$ converges and\n$$\n\\liminf_{n \\to \\infty} (a_n^2 \\sum_{k > n} \\frac{1}{a_k^2}) > 0\n$$\nis not an irrationality sequence.
\n\n [KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).
"},"Erdos264.erdos_264.variants.ko_tao_pos":{"url":"/FormalConjectures/ErdosProblems/«264»/#Erdos264___erdos_264___variants___ko_tao_pos","anchor":"Erdos264___erdos_264___variants___ko_tao_pos","docHtml":"\n On the other hand, Kovač and Tao [KoTa24] do prove that for any function $F$ with\n$\\lim_{n \\to \\infty} \\frac{F(n + 1)}{F(n)} = \\infty$ there exists such an irrationality sequence with $a_n \\sim F(n)$.
\n\n [KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).
"},"Erdos535.NoConstantPairwiseGcdCoprimeSubsets":{"url":"/FormalConjectures/ErdosProblems/«535»/#Erdos535___NoConstantPairwiseGcdCoprimeSubsets","anchor":"Erdos535___NoConstantPairwiseGcdCoprimeSubsets","docHtml":"\n No r-subset has constant pairwise GCD with coprime quotients.
\n All elements of A are positive and have exactly k prime factors,\ncounted with multiplicity.
\n Erdős [Er73] explains that Abbott pointed out the ordinary sunflower conjecture\ndoes not seem to suffice for Problem 535; the corrected stronger auxiliary\nstatement uses $Ω$, not $ω$.
"},"Erdos535.f":{"url":"/FormalConjectures/ErdosProblems/«535»/#Erdos535___f","anchor":"Erdos535___f","docHtml":"\nf r N is the maximum size of a subset A ⊆ {1,…,N} such that no r-element\nsubset of A has constant pairwise GCD.
\n Let $r \\geq 3$, and let $f_r(N)$ denote the size of the largest subset of ${1,\\ldots,N}$\nsuch that no subset of size $r$ has the same pairwise greatest common divisor between all\nelements. Erdős [Er64] proved that $f_3(N) > N^{c/\\log\\log N}$ for some constant $c > 0$, and\nconjectured this should also be an upper bound; here we state the conjectural upper bound\nfor all $r \\geq 3$.
\n\n See also [536].
"},"Erdos535.erdos_535.variants.first_open_case":{"url":"/FormalConjectures/ErdosProblems/«535»/#Erdos535___erdos_535___variants___first_open_case","anchor":"Erdos535___erdos_535___variants___first_open_case","docHtml":"\n The first open case of Erdős Problem 535 is $r = 3$: there should exist $c > 0$ such that\n$f_3(N) \\leq N^{c/\\log\\log N}$ for all sufficiently large $N$.
"},"Erdos535.erdos_535.variants.erdos_upper_bound":{"url":"/FormalConjectures/ErdosProblems/«535»/#Erdos535___erdos_535___variants___erdos_upper_bound","anchor":"Erdos535___erdos_535___variants___erdos_upper_bound","docHtml":"\n Erdős [Er64] proved that $f_r(N) \\leq N^{3/4+o(1)}$ for all $r \\geq 3$.
"},"Erdos535.erdos_535.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«535»/#Erdos535___erdos_535___variants___lower_bound","anchor":"Erdos535___erdos_535___variants___lower_bound","docHtml":"\n Erdős [Er64] proved that $f_3(N) > N^{c/\\log\\log N}$ for some constant $c > 0$.
"},"Erdos535.erdos_535.variants.abbott_hanson":{"url":"/FormalConjectures/ErdosProblems/«535»/#Erdos535___erdos_535___variants___abbott_hanson","anchor":"Erdos535___erdos_535___variants___abbott_hanson","docHtml":"\n Abbott and Hanson [AbHa70] improved Erdős's upper bound to $f_r(N) \\leq N^{1/2+o(1)}$\nfor all $r \\geq 3$.
"},"Erdos535.erdos_535.variants.sunflower_strong":{"url":"/FormalConjectures/ErdosProblems/«535»/#Erdos535___erdos_535___variants___sunflower_strong","anchor":"Erdos535___erdos_535___variants___sunflower_strong","docHtml":"\n Erdős [Er73] records that Abbott pointed out the ordinary sunflower conjecture\ndoes not seem to suffice here. The stronger auxiliary conjecture uses $Ω(n)=k$,\ni.e. prime factors counted with multiplicity; this stronger statement would imply\nthe conjectured upper bound for $f_r(N)$.
"},"Erdos535.erdos_535.variants.sunflower_erdos_rado":{"url":"/FormalConjectures/ErdosProblems/«535»/#Erdos535___erdos_535___variants___sunflower_erdos_rado","anchor":"Erdos535___erdos_535___variants___sunflower_erdos_rado","docHtml":"\n For the stronger $Ω(n)=k$ variant above, the Erdős–Rado method gives the weaker\nbound $c_r^k \\cdot k!$; see Erdős [Er73].
"},"Erdos299.erdos_299":{"url":"/FormalConjectures/ErdosProblems/«299»/#Erdos299___erdos_299","anchor":"Erdos299___erdos_299","docHtml":"\n Is there an infinite sequence $a_1 < a_2 < \\dots$ such that $a_{i+1} - a_i = O(1)$ and no finite\nsum of $\\frac{1}{a_i}$ is equal to 1?
\n\n There does not exist such a sequence, which follows from the positive solution to\n[erdosproblems.com/298] by Bloom [Bl21].
\n\n This was formalized in Lean 3 by Bloom and Mehta.
"},"Erdos299.erdos_299.variants.density":{"url":"/FormalConjectures/ErdosProblems/«299»/#Erdos299___erdos_299___variants___density","anchor":"Erdos299___erdos_299___variants___density","docHtml":"\n The corresponding question is also false if one replaces sequences such that $a_{i+1} - a_i = O(1)$\nwith sets of positive density, as follows from [Bl21].
\n\n The statement is as follows:\nIf $A \\subset \\mathbb{N}$ has positive upper density (and hence certainly if $A$ has positive\ndensity) then there is a finite $S \\subset A$ such that $\\sum_{n \\in S} \\frac{1}{n} = 1$.
"},"Erdos200.longestPrimeArithmeticProgressions":{"url":"/FormalConjectures/ErdosProblems/«200»/#Erdos200___longestPrimeArithmeticProgressions","anchor":"Erdos200___longestPrimeArithmeticProgressions","docHtml":"\n The length of the longest arithmetic progression of primes in ${1,\\ldots,n}$.
"},"Erdos200.erdos_200":{"url":"/FormalConjectures/ErdosProblems/«200»/#Erdos200___erdos_200","anchor":"Erdos200___erdos_200","docHtml":"\n Does the longest arithmetic progression of primes in ${1,\\ldots,N}$ have length $o(\\log N)$?
"},"Erdos200.erdos_200.variants.upper":{"url":"/FormalConjectures/ErdosProblems/«200»/#Erdos200___erdos_200___variants___upper","anchor":"Erdos200___erdos_200___variants___upper","docHtml":"\n It follows from the prime number theorem that such a progression has length $\\leq(1+o(1))\\log N$.
"},"Erdos847.HasFew3APs":{"url":"/FormalConjectures/ErdosProblems/«847»/#Erdos847___HasFew3APs","anchor":"Erdos847___HasFew3APs","docHtml":"\nHasFew3APs A means that $A \\subset \\mathbb{N}$ is a set for which there exists some $\\epsilon > 0$ such that\nin any subset of $A$ of size $n$ there is a subset of size at least $\\epsilon n$ which contains no\nthree-term arithmetic progression.
\n Let $A \\subset \\mathbb{N}$ be an infinite set for which there exists some $\\epsilon > 0$ such that\nin any subset of $A$ of size $n$ there is a subset of size at least $\\epsilon n$ which contains no\nthree-term arithmetic progression.
\n\n Is it true that $A$ is the union of a finite number of sets which contain no three-term arithmetic\nprogression?
\n\n A negative answer was given by Reiher, Rödl, and Sales [RRS24], who proved that, for any\n$0<\\mu<1/2$, there exists $A\\subseteq \\mathbb{N}$ such that every finite colouring of $A$ contains\na three-term arithmetic progression, and yet every subset of $A$ of size $n$ contains a subset of\nsize $\\geq \\mu n$ without a three-term arithmetic progression.
"},"Erdos499.erdos_499":{"url":"/FormalConjectures/ErdosProblems/«499»/#Erdos499___erdos_499","anchor":"Erdos499___erdos_499","docHtml":"\n Let $M$ be a real $n \\times n$ doubly stochastic matrix. Does there exist some $σ \\in S_n$ such that\n$$\n\\prod_{1 \\leq i \\leq n} M_{i, σ(i)} \\geq n^{-n}?\n$$\nThis is true, and was proved by Marcus and Minc [MaMi62]
\n\n [MaMi62] Marcus, Marvin and Minc, Henryk, Some results on doubly stochastic matrices. Proc. Amer. Math. Soc. (1962), 571-579.
"},"Erdos499.vanDerWaerden":{"url":"/FormalConjectures/ErdosProblems/«499»/#Erdos499___vanDerWaerden","anchor":"Erdos499___vanDerWaerden","docHtml":"\n The conjecture of van der Waerden, which states that the permanent of a doubly stochastic matrix is\nat least $n^{-n} n!$.
\n\n Proved by Gyires [Gy80], Egorychev [Eg81], and Falikman [Fa81].
\n\n [Gy80] Gyires, B., The common source of several inequalities concerning doubly stochastic matrices. Publ. Math. Debrecen (1980), 291-304.\n[Eg81] Egorychev, G. P., The solution of the van der Waerden problem for permanents. Dokl. Akad. Nauk SSSR (1981), 1041-1044.\n[Fa81] Falikman, D. I., Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix. Mat. Zametki (1981), 931-938, 957.
"},"Erdos499.erdos_499.variants.one_le":{"url":"/FormalConjectures/ErdosProblems/«499»/#Erdos499___erdos_499___variants___one_le","anchor":"Erdos499___erdos_499___variants___one_le","docHtml":"\n A weaker version of Erdős' problem 499, which asks whether for every doubly stochastic matrix, there\nexists a permutation $σ \\in S_n$ with $M_{i, σ(i)} ≠ 0$ and such that\n$$\n\\sum_{1 \\leq i \\leq n} M_{i, σ(i)} \\geq 1\n$$\nProved by Marcus and Ree [MaRe59].
\n\n [MaRe59] Marcus, M. and Ree, R., Diagonals of doubly stochastic matrices. Quart. J. Math. Oxford Ser. (2) (1959), 296-302.
"},"Erdos170.PerfectRuler":{"url":"/FormalConjectures/ErdosProblems/«170»/#Erdos170___PerfectRuler","anchor":"Erdos170___PerfectRuler","docHtml":"\n An $N$-perfect ruler is a finite subset $A \\subseteq \\mathbb{N}$ (the marks), such that each\npositive integer $k \\leq N$ can be measured, that is, expressed as a difference $k = a_1 - a_0$\nwith $a_0, a_1 \\in A$. The set $A$ is then also called a difference basis w.r.t. $N$.
"},"Erdos170.PerfectRulersLengthN":{"url":"/FormalConjectures/ErdosProblems/«170»/#Erdos170___PerfectRulersLengthN","anchor":"Erdos170___PerfectRulersLengthN","docHtml":"\n We define the set of all $N$-perfect rulers $A$ of length $N$, i.e.\nsubsets $A \\subseteq {0, \\dots, N}$, s.t. $A$ is $N$-perfect.\nThis is also called a restricted difference basis w.r.t. $N$.
"},"Erdos170.TrivialRuler":{"url":"/FormalConjectures/ErdosProblems/«170»/#Erdos170___TrivialRuler","anchor":"Erdos170___TrivialRuler","docHtml":"\n The trivial ruler with all marks ${0, \\dots, N}$.
"},"Erdos170.trivial_ruler_is_perfect":{"url":"/FormalConjectures/ErdosProblems/«170»/#Erdos170___trivial_ruler_is_perfect","anchor":"Erdos170___trivial_ruler_is_perfect","docHtml":"\n Sanity Check: the trivial ruler is actually a perfect ruler if $K \\geq N$
"},"Erdos170.F":{"url":"/FormalConjectures/ErdosProblems/«170»/#Erdos170___F","anchor":"Erdos170___F","docHtml":"\n We define a function F N as the minimum cardinality of an N-perfect ruler of length N.
\n The problem is to determine the limit of the sequence $\\frac{F(N)}{\\sqrt{N}}$ as $N \\to \\infty$.
"},"Erdos170.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«170»/#Erdos170___lower_bound","anchor":"Erdos170___lower_bound","docHtml":"\n A known lower bound to the limit by Leech [Le56], which is $1.56\\dots$.
"},"Erdos170.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«170»/#Erdos170___upper_bound","anchor":"Erdos170___upper_bound","docHtml":"\n A known upper bound obtained by constructing Wichmann's Rulers [Wi63].
"},"Erdos170.erdos170.existing_bounds":{"url":"/FormalConjectures/ErdosProblems/«170»/#Erdos170___erdos170___existing_bounds","anchor":"Erdos170___erdos170___existing_bounds","docHtml":"\n The existence of the limit has been proved by Erdős and Gál [ErGa48].\nThe lower bound has been proven by Leech [Le56], who refined an argument of Rédei and Rényi.\nThe upper bound is due to Wichmann [Wi63].
"},"Erdos241.f":{"url":"/FormalConjectures/ErdosProblems/«241»/#Erdos241___f","anchor":"Erdos241___f","docHtml":"\n Let $f(N)$ be the maximum size of $A\\subseteq {1,\\ldots,N}$ such that the sums $a+b+c$ with\n$a,b,c\\in A$ are all distinct (aside from the trivial coincidences).
\n\n Formalization note: this is generalized to allow for different $r$.
"},"Erdos241.erdos_241":{"url":"/FormalConjectures/ErdosProblems/«241»/#Erdos241___erdos_241","anchor":"Erdos241___erdos_241","docHtml":"\n Is it true that $f(N)\\sim N^{1/3}$?
\n\n Originally asked to Erdős by Bose.
\n\n This is discussed in problem C11 of Guy's collection [Gu04].
"},"Erdos241.erdos_241.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«241»/#Erdos241___erdos_241___variants___lower_bound","anchor":"Erdos241___erdos_241___variants___lower_bound","docHtml":"\n Bose and Chowla [BoCh62] provided a construction proving one half of this, namely\n$(1+o(1))N^{1/3}\\leq f(N)$.
"},"Erdos241.erdos_241.variants.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«241»/#Erdos241___erdos_241___variants___upper_bound","anchor":"Erdos241___erdos_241___variants___upper_bound","docHtml":"\n The best upper bound known to date is due to Green [Gr01], $f(N) \\leq ((7/2)^{1/3}+o(1))N^{1/3}$.\n(note that $(7/2)^{1/3}\\approx 1.519$).
"},"Erdos241.BoseChowlaConjecture":{"url":"/FormalConjectures/ErdosProblems/«241»/#Erdos241___BoseChowlaConjecture","anchor":"Erdos241___BoseChowlaConjecture","docHtml":"\n The conjecture that the size of the set $A\\subseteq {1,\\ldots,N}$ is asymptotically $N^{1/r}$.
"},"Erdos241.erdos_241.variants.generalization":{"url":"/FormalConjectures/ErdosProblems/«241»/#Erdos241___erdos_241___variants___generalization","anchor":"Erdos241___erdos_241___variants___generalization","docHtml":"\n More generally, Bose and Chowla [BoCh62] conjectured that the maximum size of\n$A\\subseteq {1,\\ldots,N}$ with all $r$-fold sums distinct (aside from the trivial coincidences)\nthen $\\lvert A\\rvert \\sim N^{1/r}.$
"},"Erdos241.erdos_241.variants.r_eq_2":{"url":"/FormalConjectures/ErdosProblems/«241»/#Erdos241___erdos_241___variants___r_eq_2","anchor":"Erdos241___erdos_241___variants___r_eq_2","docHtml":"\n This is known only for $r=2$ (see [erdosproblems.com/30]).
"},"Erdos727.erdos_727":{"url":"/FormalConjectures/ErdosProblems/«727»/#Erdos727___erdos_727","anchor":"Erdos727___erdos_727","docHtml":"\n Let $k ≥ 2$. Does $((n+k)!)^2∣(2n)!$ hold for infinitely many $n$?
"},"Erdos727.erdos_727.variants.k_2":{"url":"/FormalConjectures/ErdosProblems/«727»/#Erdos727___erdos_727___variants___k_2","anchor":"Erdos727___erdos_727___variants___k_2","docHtml":"\n It is open even for $k = 2$.\nLet $k = 2$. Does $((n+k)!)^2∣(2n)!$ hold for infinitely many n?
"},"Erdos727.erdos_727.variants.k_1":{"url":"/FormalConjectures/ErdosProblems/«727»/#Erdos727___erdos_727___variants___k_1","anchor":"Erdos727___erdos_727___variants___k_1","docHtml":"\n Balakran proved this holds for $k = 1$.
\n\n Let $k = 1$. Does $((n+k)!)^2∣(2n)!$ for infinitely many $n$?
"},"Erdos727.erdos_727.variants.k_1_2":{"url":"/FormalConjectures/ErdosProblems/«727»/#Erdos727___erdos_727___variants___k_1_2","anchor":"Erdos727___erdos_727___variants___k_1_2","docHtml":"\n Erdős, Graham, Ruzsa, and Straus observe that the method of Balakran can be further used to prove\nthat there are infinitely many $n$ such that $(n+k)!(n+1)!∣(2n)!$
"},"Erdos422.f":{"url":"/FormalConjectures/ErdosProblems/«422»/#Erdos422___f","anchor":"Erdos422___f","docHtml":"\n Let $f(1) = f(2) = 1$ and for $n > 2$\n$$\nf(n) = f(n - f(n - 1)) + f(n - f(n - 2)).\n$$
\n\n Note: It is not known whether $f(n)$ is well-defined for all $n$.
"},"Erdos422.erdos_422":{"url":"/FormalConjectures/ErdosProblems/«422»/#Erdos422___erdos_422","anchor":"Erdos422___erdos_422","docHtml":"\n Does $f(n)$ miss infinitely many integers?
"},"Erdos422.erdos_422.variants.surjective":{"url":"/FormalConjectures/ErdosProblems/«422»/#Erdos422___erdos_422___variants___surjective","anchor":"Erdos422___erdos_422___variants___surjective","docHtml":"\n Is $f$ surjective?
"},"Erdos422.erdos_422.variants.growth_rate":{"url":"/FormalConjectures/ErdosProblems/«422»/#Erdos422___erdos_422___variants___growth_rate","anchor":"Erdos422___erdos_422___variants___growth_rate","docHtml":"\n How does $f$ grow?
"},"Erdos422.erdos_422.variants.eventually_const":{"url":"/FormalConjectures/ErdosProblems/«422»/#Erdos422___erdos_422___variants___eventually_const","anchor":"Erdos422___erdos_422___variants___eventually_const","docHtml":"\n Does $f$ become stationary at some point?
"},"Erdos268.erdos_268":{"url":"/FormalConjectures/ErdosProblems/«268»/#Erdos268___erdos_268","anchor":"Erdos268___erdos_268","docHtml":"X be the set of points in Fin d → ℝ of the shape\nfun i : Fin d => ∑' n : A, (1 : ℝ) / (n + i) for some infinite subset A ⊆ ℕ such that\n1 / n is summable over A. X has nonempty interior. This is proved in [KoTa24].\n The set of $N$ such that any $N$ points in the plane, no three on a line,\ncontain a convex $n$-gon.
"},"Erdos107.f":{"url":"/FormalConjectures/ErdosProblems/«107»/#Erdos107___f","anchor":"Erdos107___f","docHtml":"\n The function $f(n)$ specified in erdos_107.
\n Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line,\ncontain $n$ points which form the vertices of a convex $n$-gon.\nProve that $f(n) = 2^{n-2} + 1$.
"},"Erdos107.nonempty_cardSet":{"url":"/FormalConjectures/ErdosProblems/«107»/#Erdos107___nonempty_cardSet","anchor":"Erdos107___nonempty_cardSet","docHtml":"\n For every $n ≥ 3$, there exists $N$ such that any $N$ points, no three on a line,\ncontain a convex $n$-gon.
"},"Erdos107.f_zero_eq":{"url":"/FormalConjectures/ErdosProblems/«107»/#Erdos107___f_zero_eq","anchor":"Erdos107___f_zero_eq","docHtml":"\n Depending on details of definitions,\nthe statement is false or trivial for $n < 3$.
"},"Erdos107.f_three_eq":{"url":"/FormalConjectures/ErdosProblems/«107»/#Erdos107___f_three_eq","anchor":"Erdos107___f_three_eq"},"Erdos107.variants.ersz_bounds":{"url":"/FormalConjectures/ErdosProblems/«107»/#Erdos107___variants___ersz_bounds","anchor":"Erdos107___variants___ersz_bounds","docHtml":"\n Erdős and Szekeres proved the bounds\n$$\n2^{n-2} + 1 ≤ f(n) ≤ \\binom{2n-4}{n-2} + 1\n$$\n([ErSz60] and [ErSz35] respectively).
\n\n [ErSz60] Erdős, P. and Szekeres, G.,
\n [ErSz35] Erdős, P. and Szekeres, G.,
\n Suk [Su17] proved\n$$\nf(n) ≤ 2^{(1+o(1))n}.\n$$
\n\n [Su17] Suk, Andrew,
\n The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20],\nwho prove\n$$\nf(n) ≤ 2^{n+O(\\sqrt{n\\log n})}.\n$$
\n\n [HMPT20] Holmsen, Andreas F. and Mojarrad, Hossein Nassajian and Pach, János and Tardos, Gábor,\n
\n Let $A \\subseteq (1, \\infty)$ be a countably infinite set such that for all $x\\neq y\\in A$ and\nintegers $k \\geq 1$ we have $|kx - y| \\geq 1$.
"},"Erdos143.erdos_143.parts.i":{"url":"/FormalConjectures/ErdosProblems/«143»/#Erdos143___erdos_143___parts___i","anchor":"Erdos143___erdos_143___parts___i","docHtml":"\n Does this imply that\n$$\n\\liminf \\frac{|A \\cap [1,x]|}{x} = 0?\n$$
"},"Erdos143.erdos_143.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«143»/#Erdos143___erdos_143___parts___ii","anchor":"Erdos143___erdos_143___parts___ii","docHtml":"\n Or\n$$\n\\sum_{x \\in A} \\frac{1}{x \\log x} < \\infty,\n$$
"},"Erdos56.WeaklyDivisible":{"url":"/FormalConjectures/ErdosProblems/«56»/#Erdos56___WeaklyDivisible","anchor":"Erdos56___WeaklyDivisible","docHtml":"\n Say a set of natural numbers is k-weakly divisible if any k+1 elements\nof A are not relatively prime.
\n A singleton is k-weakly divisble if k ≠ 0.
\n No non-empty set is 1-weakly divisible.
\nMaxWeaklyDivisible N k is the size of the largest k-weakly divisible subset of {1,..., N}
\nFirstPrimesMultiples N k is the set of numbers in {1,..., N} that are\na multiple of one of the first k primes.
\n An example of a k-weakly divisible set is the subset of {1, ..., N}\ncontaining the multiples of the first k primes.
\n Suppose $A \\subseteq {1,\\dots,N}$ is such that there are no $k+1$ elements of $A$ which are\nrelatively prime. An example is the set of all multiples of the first $k$ primes.\nIs this the largest such set? To avoid trivial counterexamples, we must insist that $N$ be at\nleast the $k$th prime.
"},"Erdos16.Erdos16Set":{"url":"/FormalConjectures/ErdosProblems/«16»/#Erdos16___Erdos16Set","anchor":"Erdos16___Erdos16Set","docHtml":"\n The set of odd integers not of the form $2^k+p$.
"},"Erdos16.density_zero":{"url":"/FormalConjectures/ErdosProblems/«16»/#Erdos16___density_zero","anchor":"Erdos16___density_zero","docHtml":"\n A set of natural numbers has density 0.
"},"Erdos16.erdos_16":{"url":"/FormalConjectures/ErdosProblems/«16»/#Erdos16___erdos_16","anchor":"Erdos16___erdos_16","docHtml":"\n Is the set of odd integers not of the form $2^k+p$ the union of an infinite arithmetic progression\nand a set of density $0$?
\n\n Erdős called this conjecture \"rather silly\".
\n\n Chen [Ch23] has proved the answer is no.
\n\n This was formalized in Lean by Chin using Aristotle.
"},"Erdos16.positive_lower_density":{"url":"/FormalConjectures/ErdosProblems/«16»/#Erdos16___positive_lower_density","anchor":"Erdos16___positive_lower_density","docHtml":"\n A set of natural numbers has positive lower density.
"},"Erdos16.erdos_16.variant.romanoff":{"url":"/FormalConjectures/ErdosProblems/«16»/#Erdos16___erdos_16___variant___romanoff","anchor":"Erdos16___erdos_16___variant___romanoff","docHtml":"\n Romanoff [Ro34] showed that the set of odd integers of this form has positive density.
"},"Erdos16.erdos_16.variant.erdos":{"url":"/FormalConjectures/ErdosProblems/«16»/#Erdos16___erdos_16___variant___erdos","anchor":"Erdos16___erdos_16___variant___erdos","docHtml":"\n Using covering congruences Erdős [Er50] proved that the set of odd integers which are not of this\nform contains an infinite arithmetic progression.
"},"Erdos940.erdos_940":{"url":"/FormalConjectures/ErdosProblems/«940»/#Erdos940___erdos_940","anchor":"Erdos940___erdos_940","docHtml":"\n Let $r \\ge 3$. Is it true that the set of integers which are the sum of at most $r$ $r$-powerful numbers\nhas density $0$?
"},"Erdos940.erdos_940.variants.two":{"url":"/FormalConjectures/ErdosProblems/«940»/#Erdos940___erdos_940___variants___two","anchor":"Erdos940___erdos_940___variants___two","docHtml":"\n The set of integers which are the sum of at most two $2$-powerful numbers has density $0$.
"},"Erdos940.erdos_940.variants.three_cubes":{"url":"/FormalConjectures/ErdosProblems/«940»/#Erdos940___erdos_940___variants___three_cubes","anchor":"Erdos940___erdos_940___variants___three_cubes","docHtml":"\n Is it true that the set of integers which are the sum of at most three cubes has density $0$?
"},"Erdos940.erdos_940.variants.large_integers":{"url":"/FormalConjectures/ErdosProblems/«940»/#Erdos940___erdos_940___variants___large_integers","anchor":"Erdos940___erdos_940___variants___large_integers","docHtml":"\n It is not known if all large integers are the sum of at most $r$-many $r$-powerful numbers.
"},"Erdos940.erdos_940.variants.three_powerful":{"url":"/FormalConjectures/ErdosProblems/«940»/#Erdos940___erdos_940___variants___three_powerful","anchor":"Erdos940___erdos_940___variants___three_powerful","docHtml":"\n Heath-Brown [He88] has proved that all large numbers are the sum of at most three\n$2$-powerful numbers.
\n\n [He88] Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. (1988), 137--163.
"},"Erdos681.IsLPF":{"url":"/FormalConjectures/ErdosProblems/«681»/#Erdos681___IsLPF","anchor":"Erdos681___IsLPF","docHtml":"\nIsLPF p m says that p is the least prime factor of m.
\nErdős problem 681.\nIs it true that for all large $n$ there exists $k$\nsuch that $n + k$ is composite and $p(n+k) > k^2$,\nwhere $p(m)$ is the least prime factor of $m$ ?
"},"Erdos672.Erdos672With":{"url":"/FormalConjectures/ErdosProblems/«672»/#Erdos672___Erdos672With","anchor":"Erdos672___Erdos672With","docHtml":"\n Erdős problem 672 conjectures that the below holds for any $k ≥ 4$ and $l > 1$.
"},"Erdos672.erdos_672":{"url":"/FormalConjectures/ErdosProblems/«672»/#Erdos672___erdos_672","anchor":"Erdos672___erdos_672","docHtml":"\n Can the product of an arithmetic progression of positive integers $n, n + d, ..., n + (k - 1)d$\nof length ≥ 4, with $(n, d) = 1$, be a perfect power?
"},"Erdos672.erdos_672.variants.euler":{"url":"/FormalConjectures/ErdosProblems/«672»/#Erdos672___erdos_672___variants___euler","anchor":"Erdos672___erdos_672___variants___euler","docHtml":"\n According to https://www.erdosproblems.com/672, Euler proved this.
"},"Erdos672.erdos_672.variants.oblath":{"url":"/FormalConjectures/ErdosProblems/«672»/#Erdos672___erdos_672___variants___oblath","anchor":"Erdos672___erdos_672___variants___oblath","docHtml":"\n According to https://www.erdosproblems.com/672, Obláth proved this.
\n\n [Ob51] Oblath, Richard, Eine Bemerkung über Produkte aufeinander folgender Zahlen.\nJ. Indian Math. Soc. (N.S.) (1951), 135-139.
"},"Erdos1084.f":{"url":"/FormalConjectures/ErdosProblems/«1084»/#Erdos1084___f","anchor":"Erdos1084___f","docHtml":"\n The maximal number of pairs of points which are distance 1 apart that a set of n 1-separated\npoints in ℝ^d make.
\n It is easy to check that $f_1(n) = n - 1$.
"},"Erdos1084.erdos_1084.variants.easy_upper_d2":{"url":"/FormalConjectures/ErdosProblems/«1084»/#Erdos1084___erdos_1084___variants___easy_upper_d2","anchor":"Erdos1084___erdos_1084___variants___easy_upper_d2","docHtml":"\n It is easy to check that $f_2(n) < 3n$.
"},"Erdos1084.erdos_1084.variants.upper_d2":{"url":"/FormalConjectures/ErdosProblems/«1084»/#Erdos1084___erdos_1084___variants___upper_d2","anchor":"Erdos1084___erdos_1084___variants___upper_d2","docHtml":"\n Erdős showed that there is some constant $c > 0$ such that $f_2(n) < 3n - c n^{1/2}$.
"},"Erdos1084.erdos_1084.variants.triangular_optimal_d2":{"url":"/FormalConjectures/ErdosProblems/«1084»/#Erdos1084___erdos_1084___variants___triangular_optimal_d2","anchor":"Erdos1084___erdos_1084___variants___triangular_optimal_d2","docHtml":"\n Erdős conjectured that the triangular lattice is best possible in 2D, in particular that\n$f_2(3n^2 + 3n + 1) < 9n^2 + 3n$.
\n\n Note: in [Er75f] is read $9n^2 + 6n$, but this seems to be a typo.
"},"Erdos1084.erdos_1084.variants.upper_lower_d3":{"url":"/FormalConjectures/ErdosProblems/«1084»/#Erdos1084___erdos_1084___variants___upper_lower_d3","anchor":"Erdos1084___erdos_1084___variants___upper_lower_d3","docHtml":"\n Erdős claims the existence of two constants $c_1, c_2 > 0$\nsuch that $6n - c_1 n^{2/3} ≤ f_3(n) \\le 6n - c_2 n^{2/3}$.
"},"Erdos13.IsForbiddenTripleFree":{"url":"/FormalConjectures/ErdosProblems/«13»/#Erdos13___IsForbiddenTripleFree","anchor":"Erdos13___IsForbiddenTripleFree","docHtml":"\n A finite set of naturals A is said to be forbidden-triple-free if for all a, b, c ∈ A,\nif a < min(b, c) then a does not divide b + c.
\n If $A \\subseteq {1, ..., N}$ is a set with no $a, b, c \\in A$ such that $a | (b+c)$ and\n$a < \\min(b,c)$, then $|A| \\le N/3 + O(1)$. This has been solved by Bedert [Be23].
\n\n [Be23] Bedert, B.,
\n A general version asks, for a fixed $r \\in \\mathbb{N}$, if a set\n$A \\subseteq {1, ..., N}$ has no $a \\in A$ and $b_1, ..., b_r \\in A$ such that\n$a | (b_1 + ... + b_r)$ and $a < \\min(b_1, ..., b_r)$, then is it true that\n$|A| \\le N/(r+1) + O(1)$?
"},"Erdos1074.EHSNumbers":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___EHSNumbers","anchor":"Erdos1074___EHSNumbers","docHtml":"\n The EHS numbers (after Erdős, Hardy, and Subbarao) are those $m\\geq 1$ such that there\nexists a prime $p\\not\\equiv 1\\pmod{m}$ such that $m! + 1 \\equiv 0\\pmod{p}$.
"},"Erdos1074.PillaiPrimes":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___PillaiPrimes","anchor":"Erdos1074___PillaiPrimes","docHtml":"\n The Pillai primes are those primes $p$ such that there exists an $m \\ge 1$ with\n$p\\not\\equiv 1\\pmod{m}$ such that $m! + 1 \\equiv 0\\pmod{p}$
"},"Erdos1074.two_not_mem_pillaiPrimes":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___two_not_mem_pillaiPrimes","anchor":"Erdos1074___two_not_mem_pillaiPrimes"},"Erdos1074.twentyThree_mem_pillaiPrimes":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___twentyThree_mem_pillaiPrimes","anchor":"Erdos1074___twentyThree_mem_pillaiPrimes"},"Erdos1074.erdos_1074.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___erdos_1074___parts___i","anchor":"Erdos1074___erdos_1074___parts___i","docHtml":"\n Let $S$ be the set of all $m\\geq 1$ such that there exists a prime $p\\not\\equiv 1\\pmod{m}$ such\nthat $m! + 1 \\equiv 0\\pmod{p}$. Does\n$$\n\\lim\\frac{|S\\cap[1, x]|}{x}\n$$\nexist?
"},"Erdos1074.erdos_1074.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___erdos_1074___parts___ii","anchor":"Erdos1074___erdos_1074___parts___ii","docHtml":"\n Let $S$ be the set of all $m\\geq 1$ such that there exists a prime $p\\not\\equiv 1\\pmod{m}$ such\nthat $m! + 1 \\equiv 0\\pmod{p}$. What is\n$$\n\\lim\\frac{|S\\cap[1, x]|}{x}?\n$$
"},"Erdos1074.erdos_1074.parts.iii":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___erdos_1074___parts___iii","anchor":"Erdos1074___erdos_1074___parts___iii","docHtml":"\n Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with\n$p\\not\\equiv 1\\pmod{m}$ such that $m! + 1 \\equiv 0\\pmod{p}$, then does\n$$\n\\lim\\frac{|P\\cap[1, x]|}{\\pi(x)}\n$$\nexist?
"},"Erdos1074.erdos_1074.parts.iv":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___erdos_1074___parts___iv","anchor":"Erdos1074___erdos_1074___parts___iv","docHtml":"\n Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with\n$p\\not\\equiv 1\\pmod{m}$ such that $m! + 1 \\equiv 0\\pmod{p}$, then what is\n$$\n\\lim\\frac{|P\\cap[1, x]|}{\\pi(x)}?\n$$
"},"Erdos1074.erdos_1074.variants.mem_pillaiPrimes":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___erdos_1074___variants___mem_pillaiPrimes","anchor":"Erdos1074___erdos_1074___variants___mem_pillaiPrimes","docHtml":"\n Pillai [Pi30] raised the question of whether there exist any primes in $P$. This was answered\nby Chowla, who noted that, for example, $14! + 1 \\equiv 18! + 1 \\equiv 0 \\pmod{23}$.
"},"Erdos1074.erdos_1074.variants.EHSNumbers_infinite":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___erdos_1074___variants___EHSNumbers_infinite","anchor":"Erdos1074___erdos_1074___variants___EHSNumbers_infinite","docHtml":"\n Erdős, Hardy, and Subbarao proved that $S$ is infinite.
\n\n Formal proof linked here provided by AlphaProof.
"},"Erdos1074.erdos_1074.variants.PillaiPrimes_infinite":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___erdos_1074___variants___PillaiPrimes_infinite","anchor":"Erdos1074___erdos_1074___variants___PillaiPrimes_infinite","docHtml":"\n Erdős, Hardy, and Subbarao proved that $P$ is infinite.
"},"Erdos1074.erdos_1074.variants.EHSNumbers_init":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___erdos_1074___variants___EHSNumbers_init","anchor":"Erdos1074___erdos_1074___variants___EHSNumbers_init","docHtml":"\n The sequence $S$ begins $8, 9, 13, 14, 15, 16, 17, ...$
"},"Erdos1074.erdos_1074.variants.PillaiPrimes_init":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___erdos_1074___variants___PillaiPrimes_init","anchor":"Erdos1074___erdos_1074___variants___PillaiPrimes_init","docHtml":"\n The sequence $P$ begins $23, 29, 59, 61, 67, 71, ...$
"},"Erdos1074.erdos_1074.variants.EHSNumbers_one_half":{"url":"/FormalConjectures/ErdosProblems/«1074»/#Erdos1074___erdos_1074___variants___EHSNumbers_one_half","anchor":"Erdos1074___erdos_1074___variants___EHSNumbers_one_half","docHtml":"\n Regarding the first question, Hardy and Subbarao computed all EHS numbers up to $2^{10}$, and\nwrite \"...if this trend conditions we expect [the limit] to be around 0.5, if it exists.\"
"},"Erdos1105.erdos_1105.parts.i":{"url":"/FormalConjectures/ErdosProblems/«1105»/#Erdos1105___erdos_1105___parts___i","anchor":"Erdos1105___erdos_1105___parts___i","docHtml":"\n The anti-Ramsey number $\\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the\nedges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all edges\nhave different colours).
\n\n Let $C_k$ be the cycle on $k$ vertices. Is it true that\n$\\mathrm{AR}(n,C_k)=\\left(\\frac{k-2}{2}+\\frac{1}{k-1}\\right)n+O(1)$?
\n\n Montellano-Ballesteros and Neumann-Lara [MoNe05] gave an exact formula for $\\mathrm{AR}(n,C_k)$,\nwhich implies in particular that\n$\\mathrm{AR}(n,C_k)=\\left(\\frac{k-2}{2}+\\frac{1}{k-1}\\right)n+O(1).$
"},"Erdos1105.erdos_1105.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«1105»/#Erdos1105___erdos_1105___parts___ii","anchor":"Erdos1105___erdos_1105___parts___ii","docHtml":"\n Let $P_k$ be the path on $k$ vertices and $\\ell=\\lfloor\\frac{k-1}{2}\\rfloor$. If $n\\geq k\\geq 5$\nthen is $\\mathrm{AR}(n,P_k)$ equal to $\\max\\left(\\binom{k-2}{2}+1,\n\\binom{\\ell-1}{2}+(\\ell-1)(n-\\ell+1)+\\epsilon\\right)$where $\\epsilon=1$ if $k$ is odd and\n$\\epsilon=2$ otherwise?
\n\n A proof of the formula for $\\mathrm{AR}(n,P_k)$ for all $n\\geq k\\geq 5$ has been announced by\nYuan [Yu21].
"},"Erdos541.erdos_541":{"url":"/FormalConjectures/ErdosProblems/«541»/#Erdos541___erdos_541","anchor":"Erdos541___erdos_541","docHtml":"\n Let $a_1, \\dots, a_p$ be (not necessarily distinct) residues modulo a prime $p$, such that there\nexists some $r$ so that if $S \\subseteq [p]$ is non-empty and\n$$\\sum_{i \\in S} a_i \\equiv 0 \\pmod{p}$$\nthen $|S| = r$.
\n\n Must there be at most two distinct residues amongst the $a_i$?
\n\n This was formalized in Lean by Alexeev using Aristotle and ChatGPT.
"},"Erdos541.erdos_541.variants.general_moduli":{"url":"/FormalConjectures/ErdosProblems/«541»/#Erdos541___erdos_541___variants___general_moduli","anchor":"Erdos541___erdos_541___variants___general_moduli","docHtml":"\n Gao, Hamidoune, and Wang [GHW10] solved this for all moduli p (not necessarily prime).
\n This was proved by Erdős and Szemerédi [ErSz76] for p sufficiently large.
"},"Erdos1097.CommonDifferencesThreeTermAP":{"url":"/FormalConjectures/ErdosProblems/«1097»/#Erdos1097___CommonDifferencesThreeTermAP","anchor":"Erdos1097___CommonDifferencesThreeTermAP","docHtml":"\n Given a finite set of integers A (modelled as a Finset ℤ), the set\nCommonDifferencesThreeTermAP A consists of all integers d such that there\nis a non-trivial three-term arithmetic progression a, b, c ∈ A with\nb - a = d and c - b = d.
\n The main conjecture: for any finite set of integers $A$ with $|A| = n$, the number of distinct\ncommon differences in three-term arithmetic progressions is $O(n^{3/2})$.
\n\n This conjecture was resolved negatively by showing that the problem is exactly equivalent to\nBourgain's sums-differences question [Bo99], which was introduced as an arithmetic path towards\nthe Kakeya conjecture. Under this equivalence:
\n\n The greatest achievable exponent for this problem is equal to the smallest constant $c$\nachievable for Bourgain's sums-differences question:\n$$|A -_G B| \\ll \\max(|A|, |B|, |A +_G B|)^c$$
\n\n The $O(n^{3/2})$ prediction is disproved because the lower bound has been shown to satisfy\n$c \\ge 1.77898$ (due to Zheng and AlphaEvolve [GGTW25], improving on Lemm [Le15]), which is\nstrictly greater than $3/2 = 1.5$.
\n\n The best known upper bound is $c \\le 11/6 \\approx 1.833$ (due to Katz and Tao [KaTa99]).
\n\n While the specific $O(n^{3/2})$ prediction is resolved negatively, the general question of\ndetermining the exact optimal exponent $c$ remains open.
\n\n A weaker bound has been proven: there are always at most $n^2$ such values of $d$.
"},"Erdos1097.erdos_1097.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«1097»/#Erdos1097___erdos_1097___variants___lower_bound","anchor":"Erdos1097___erdos_1097___variants___lower_bound","docHtml":"\n A trivial lower bound: for sufficiently large n there exist sets $A$ with $|A| = n$ that contain at least $\\Omega(n)$\ndistinct common differences of three-term arithmetic progressions.
\n For integer $n \\geq 1$ we define the factor difference set of $n$ by\n$D(n) = {|a-b| : n=ab}$.
"},"Erdos885.erdos_885":{"url":"/FormalConjectures/ErdosProblems/«885»/#Erdos885___erdos_885","anchor":"Erdos885___erdos_885","docHtml":"\n Is it true that, for every $k \\geq 1$, there exist integers $N_1 < \\dots < N_k$ such that\n$|\\cap_i D(N_i)| \\geq k$?
"},"Erdos885.erdos_885.variants.k_eq_2":{"url":"/FormalConjectures/ErdosProblems/«885»/#Erdos885___erdos_885___variants___k_eq_2","anchor":"Erdos885___erdos_885___variants___k_eq_2","docHtml":"\n Erdős and Rosenfeld [ErRo97] proved this is true for $k=2$.
"},"Erdos885.erdos_885.variants.k_eq_3":{"url":"/FormalConjectures/ErdosProblems/«885»/#Erdos885___erdos_885___variants___k_eq_3","anchor":"Erdos885___erdos_885___variants___k_eq_3","docHtml":"\n Jiménez-Urroz [Ji99] proved this for $k=3$.
"},"Erdos885.erdos_885.variants.k_eq_4":{"url":"/FormalConjectures/ErdosProblems/«885»/#Erdos885___erdos_885___variants___k_eq_4","anchor":"Erdos885___erdos_885___variants___k_eq_4","docHtml":"\n Bremner [Br19] proved this for $k=4$.
"},"Erdos32.IsAdditiveComplementToPrimes":{"url":"/FormalConjectures/ErdosProblems/«32»/#Erdos32___IsAdditiveComplementToPrimes","anchor":"Erdos32___IsAdditiveComplementToPrimes","docHtml":"\n A set $A \\subseteq \\mathbb{N}$ is an
\n Erdős proved in [Erd54] that there exists an additive complement $A$ to the primes with\n$|A \\cap {1, \\ldots, N}| = O((\\log N)^2)$.
"},"Erdos32.erdos_32.variants.liminf_gt_one":{"url":"/FormalConjectures/ErdosProblems/«32»/#Erdos32___erdos_32___variants___liminf_gt_one","anchor":"Erdos32___erdos_32___variants___liminf_gt_one","docHtml":"\n Must every additive complement $A$ to the primes satisfy\n$\\liminf_{N \\to \\infty} \\frac{|A \\cap {1, \\ldots, N}|}{\\log N} > 1$?
"},"Erdos32.erdos_32":{"url":"/FormalConjectures/ErdosProblems/«32»/#Erdos32___erdos_32","anchor":"Erdos32___erdos_32","docHtml":"\n Does there exist a set $A \\subseteq \\mathbb{N}$ such that $|A \\cap {1, \\ldots, N}| = o((\\log N)^2)$\nand every sufficiently large integer can be written as $p + a$ for some prime $p$ and $a \\in A$?
"},"Erdos32.erdos_32.variants.log_bound":{"url":"/FormalConjectures/ErdosProblems/«32»/#Erdos32___erdos_32___variants___log_bound","anchor":"Erdos32___erdos_32___variants___log_bound","docHtml":"\n Can the bound $O(\\log N)$ be achieved for an additive complement to the primes? [Guy04] writes\nthat Erdős offered $50 for the solution.
"},"Erdos32.erdos_32.variants.ruzsa":{"url":"/FormalConjectures/ErdosProblems/«32»/#Erdos32___erdos_32___variants___ruzsa","anchor":"Erdos32___erdos_32___variants___ruzsa","docHtml":"\n Ruzsa proved that any additive complement $A$ to the primes must satisfy\n$\\liminf_{N \\to \\infty} \\frac{|A \\cap {1, \\ldots, N}|}{\\log N} \\geq e^\\gamma$,\nwhere $\\gamma$ is the Euler-Mascheroni constant.
"},"Erdos128.erdos_128":{"url":"/FormalConjectures/ErdosProblems/«128»/#Erdos128___erdos_128","anchor":"Erdos128___erdos_128","docHtml":"\n Let G be a graph with n vertices such that every subgraph on ≥ $n/2$\nvertices has more than $n^2/50$ edges. Must G contain a triangle?
"},"Erdos830.A":{"url":"/FormalConjectures/ErdosProblems/«830»/#Erdos830___A","anchor":"Erdos830___A","docHtml":"\n Let $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$.
"},"Erdos830.erdos_830.parts.i":{"url":"/FormalConjectures/ErdosProblems/«830»/#Erdos830___erdos_830___parts___i","anchor":"Erdos830___erdos_830___parts___i","docHtml":"\nErdos Problem 830, Part 1\nWe say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$. Are there\ninfinitely many amicable pairs?
"},"Erdos830.erdos_830.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«830»/#Erdos830___erdos_830___parts___ii","anchor":"Erdos830___erdos_830___parts___ii","docHtml":"\nErdos Problem 830, Part 2\nWe say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$.\nIf $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then is it true that\n$$A(x) > x^{1-o(1)}?$$
"},"Erdos830.erdos_830.variants.erdos":{"url":"/FormalConjectures/ErdosProblems/«830»/#Erdos830___erdos_830___variants___erdos","anchor":"Erdos830___erdos_830___variants___erdos","docHtml":"\n We say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$.\nIf $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then one can show that $A(x) = o(x)$.
"},"Erdos830.erdos_830.variants.pomerance":{"url":"/FormalConjectures/ErdosProblems/«830»/#Erdos830___erdos_830___variants___pomerance","anchor":"Erdos830___erdos_830___variants___pomerance","docHtml":"\n We say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$.\nIf $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then one can show that\n$A(x) \\leq x \\exp(-(\\log x)^{1/3})$.
"},"Erdos830.erdos_830.variants.pomerance_stronger":{"url":"/FormalConjectures/ErdosProblems/«830»/#Erdos830___erdos_830___variants___pomerance_stronger","anchor":"Erdos830___erdos_830___variants___pomerance_stronger","docHtml":"\n We say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$.\nIf $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then one can show that\n$A(x) \\leq x \\exp(-(\\tfrac{1}{2}+o(1))(\\log x\\log\\log x)^{1/2})$.
"},"Erdos757.IsAdmissible":{"url":"/FormalConjectures/ErdosProblems/«757»/#Erdos757___IsAdmissible","anchor":"Erdos757___IsAdmissible","docHtml":"\n We say that c is admissible if for any finit set A such that for any subset B of size 4,\n(B - B).card = 11, there exists a Sidon subset S of size at least c * A.ncard.
\n What is the supremum of the set of admissible numbers?
"},"Erdos757.erdos_757.variants.lowerBound":{"url":"/FormalConjectures/ErdosProblems/«757»/#Erdos757___erdos_757___variants___lowerBound","anchor":"Erdos757___erdos_757___variants___lowerBound","docHtml":"\n The supremum is strictly larger than 1 / 2, which is proved in [GyLe95].
\n In [GyLe95], the authors also prove that the supremum is smaller than 3 / 5.
\n Let $q_1 < q_2 < \\cdots$ be a sequence of primes such that $q_{i + 1} \\equiv 1 \\pmod{q_i}$. Is it\ntrue that\n$$\n\\lim_{k \\to \\infty} q_k^{1/k} = \\infty?\n$$
"},"Erdos695.erdos_695.variants.upperBound":{"url":"/FormalConjectures/ErdosProblems/«695»/#Erdos695___erdos_695___variants___upperBound","anchor":"Erdos695___erdos_695___variants___upperBound","docHtml":"\n Is there a sequence of primes $q_1 < q_2 < \\cdots$ such that $q_{i + 1} \\equiv 1 \\pmod{q_i}$ and\n$$\nq(k) \\leq \\exp(k (\\log k)^{1 + o(1)})?\n$$
"},"Erdos749.erdos_749":{"url":"/FormalConjectures/ErdosProblems/«749»/#Erdos749___erdos_749","anchor":"Erdos749___erdos_749","docHtml":"\n Let $\\epsilon>0$. Does there exist $A\\subseteq \\mathbb{N}$\nsuch that the lower density of $A+A$ is at least $1-\\epsilon$\nand yet $1_A\\ast 1_A(n) \\ll_\\epsilon 1$ for all $n$?
"},"Erdos623.erdos_623":{"url":"/FormalConjectures/ErdosProblems/«623»/#Erdos623___erdos_623","anchor":"Erdos623___erdos_623","docHtml":"\n Let $X$ be a set of cardinality $\\aleph_\\omega$ and $f$ be a function from the finite subsets of\n$X$ to $X$ such that $f(A)\\not\\in A$ for all $A$. Must there exist an infinite $Y\\subseteq X$\nthat is independent - that is, for all finite $B\\subset Y$ we have $f(B)\\not\\in Y$?
"},"Erdos488.erdos_488":{"url":"/FormalConjectures/ErdosProblems/«488»/#Erdos488___erdos_488","anchor":"Erdos488___erdos_488","docHtml":"\n Let $A$ be a finite set and\n$$B={ n \\geq 1 : a\\mid n\\textrm{ for some }a\\in A}.$$\nIs it true that, for every $m>n\\geq \\max(A)$,\n$$\\frac{\\lvert B\\cap [1,m]\\rvert }{m}< 2\\frac{\\lvert B\\cap [1,n]\\rvert}{n}?$$
"},"Erdos172.erdos_172":{"url":"/FormalConjectures/ErdosProblems/«172»/#Erdos172___erdos_172","anchor":"Erdos172___erdos_172","docHtml":"\n Is it true that in any finite colouring of $\\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums\nand products of distinct elements in $A$ are the same colour?
"},"Erdos694.erdos_694":{"url":"/FormalConjectures/ErdosProblems/«694»/#Erdos694___erdos_694","anchor":"Erdos694___erdos_694","docHtml":"\n Let $f_\\max(n)$ be the largest $m$ such that $\\phi(m) = n$, and\n$f_\\min(n)$ be the smallest such $m$, where $\\phi$ is Euler's\ntotient function. Investigate\n$$\n\\max_{n\\leq x}\\frac{f_\\max(n)}{f_\\min(n)}.\n$$
\n\n GPT-5.5 Pro (prompted by Price) has proved (see also the comments for a summary) that\n$$\n\\max_{n\\leq x}\\frac{f_{\\max}(n)}{f_{\\min}(n)}=(e^\\gamma+o(1))\\log\\log x.\n$$
\n\n A Lean formalisation of the reduction exists, conditional on Mertens' product theorem and\nLinnik's theorem; see the\nformal proof.
"},"Erdos694.erdos_694.variants.carmichael":{"url":"/FormalConjectures/ErdosProblems/«694»/#Erdos694___erdos_694___variants___carmichael","anchor":"Erdos694___erdos_694___variants___carmichael","docHtml":"\n Carmichael has asked whether there is an integer $n$ for which $\\phi(m) = n$ has\nexactly one solution, that is $\\frac{f_\\max(n)}{f_\\min(n)} = 1$.
"},"Erdos694.erdos_694.variants.inf_unique":{"url":"/FormalConjectures/ErdosProblems/«694»/#Erdos694___erdos_694___variants___inf_unique","anchor":"Erdos694___erdos_694___variants___inf_unique","docHtml":"\n Erdős has proved that if there exists an integer $n$ for which $\\phi(m) = n$ has\nexactly one solution, then there must be infinitely many such $n$.
"},"Erdos67.erdos_67":{"url":"/FormalConjectures/ErdosProblems/«67»/#Erdos67___erdos_67","anchor":"Erdos67___erdos_67","docHtml":"\nThe Erdős discrepancy problem
\n\n If $f\\colon \\mathbb N \\rightarrow {-1, +1}$ then is it true that for every $C>0$ there\nexist $d, m \\ge 1$ such that $$\\left\\lvert \\sum_{1\\leq k\\leq m}f(kd)\\right\\rvert > C?$$\nThis is true, and was proved by Tao [Ta16]
"},"Erdos67.erdos_67.variants.complex":{"url":"/FormalConjectures/ErdosProblems/«67»/#Erdos67___erdos_67___variants___complex","anchor":"Erdos67___erdos_67___variants___complex","docHtml":"\nThe Erdős discrepancy problem (complex variant)
\n\n If $f\\colon \\mathbb N \\rightarrow S^1 ⊆ ℂ$ then is it true that for every $C>0$ there\nexist $d, m \\ge 1$ such that $$\\left\\lvert \\sum_{1\\leq k\\leq m}f(kd)\\right\\rvert > C?$$\nThis is true, and was proved by Tao [Ta16]
"},"Erdos728.erdos_728":{"url":"/FormalConjectures/ErdosProblems/«728»/#Erdos728___erdos_728","anchor":"Erdos728___erdos_728","docHtml":"\n Let $\\varepsilon$ be sufficiently small and $C, C' > 0$. Are there integers $a, b, n$ such that\n$$a, b > \\varepsilon n\\quad a!, b! \\mid n!, (a + b - n)!, $$\nand\n$$C \\log n < a + b - n < C' \\log n ?$$
\n\n Note that the website currently displays a simpler (trivial) version of this problem because\n$a + b$ isn't assumed to be in the $n + O(\\log n)$ regime.
\n\n Barreto and ChatGPT-5.2 have proved that, for any $0 < C_1 < C_2$, there are infinitely many\n$a, b, n$ with $b = n/2$, $a = n/2 + O(\\log n)$, and $C_1 \\log n < a + b - n < C_2 \\log n$ such\nthat $a! b! \\mid n! (a + b - n)!$
\n\n This appears to answer the question in the spirit it was intended.
\n\n This was formalized in Lean by Alexeev using Aristotle.
"},"Erdos254.IsSumOfDistinct":{"url":"/FormalConjectures/ErdosProblems/«254»/#Erdos254___IsSumOfDistinct","anchor":"Erdos254___IsSumOfDistinct","docHtml":"\n An integer n can be written as a sum of distinct elements of A.
\n Let $A\\subseteq \\mathbb{N}$ be such that $\\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert \\to\n\\infty\\textrm{ as }x\\to \\infty$ and $\\sum_{n\\in A} { \\theta n}=\\infty$ for every $\\theta\\in\n(0,1)$, where ${x}$ is the distance of $x$ from the nearest integer. Then every sufficiently large\ninteger is the sum of distinct elements of $A$.
"},"Erdos254.erdos_254.variants.cassels":{"url":"/FormalConjectures/ErdosProblems/«254»/#Erdos254___erdos_254___variants___cassels","anchor":"Erdos254___erdos_254___variants___cassels","docHtml":"\n Cassels [Ca60] proved this under the alternative hypotheses $\\lim \\frac{\\lvert A\\cap [1,2x]\\rvert\n-\\lvert A\\cap [1,x]\\rvert}{\\log\\log x}=\\infty$ and $\\sum_{n\\in A} { \\theta n}^2=\\infty$ for every\n$\\theta\\in (0,1)$.
"},"Erdos184.IsCycleOrEdge":{"url":"/FormalConjectures/ErdosProblems/«184»/#Erdos184___IsCycleOrEdge","anchor":"Erdos184___IsCycleOrEdge","docHtml":"\n A graph $H$ is a cycle or an edge if it is connected and 2-regular, or if it has exactly one edge.
"},"Erdos184.IsDecomposition":{"url":"/FormalConjectures/ErdosProblems/«184»/#Erdos184___IsDecomposition","anchor":"Erdos184___IsDecomposition","docHtml":"\n D is a decomposition of G into subgraphs.
"},"Erdos184.erdos_184":{"url":"/FormalConjectures/ErdosProblems/«184»/#Erdos184___erdos_184","anchor":"Erdos184___erdos_184","docHtml":"\n Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges.
"},"Erdos184.erdos_184.variants.n_log_n":{"url":"/FormalConjectures/ErdosProblems/«184»/#Erdos184___erdos_184___variants___n_log_n","anchor":"Erdos184___erdos_184___variants___n_log_n","docHtml":"\n Erdős and Gallai [EGP66] proved that $O(n \\log n)$ many cycles and edges suffices.
"},"Erdos184.erdos_184.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«184»/#Erdos184___erdos_184___variants___lower_bound","anchor":"Erdos184___erdos_184___variants___lower_bound","docHtml":"\n The graph $K_{3,n-3}$ shows that at least $(1+c)n$ many cycles and edges are required, for some\nconstant $c>0$.
"},"Erdos184.erdos_184.variants.covering":{"url":"/FormalConjectures/ErdosProblems/«184»/#Erdos184___erdos_184___variants___covering","anchor":"Erdos184___erdos_184___variants___covering","docHtml":"\n In [Er71] Erdős suggests that only $n-1$ many cycles and edges are required if we do not\nrequire them to be edge-disjoint.
"},"Erdos184.erdos_184.variants.bucic_montgomery":{"url":"/FormalConjectures/ErdosProblems/«184»/#Erdos184___erdos_184___variants___bucic_montgomery","anchor":"Erdos184___erdos_184___variants___bucic_montgomery","docHtml":"\n The best bound available is due to Bucić and Montgomery [BM22], who prove that $O(n\\log^* n)$ many\ncycles and edges suffice, where $\\log^*$ is the iterated logarithm function.
"},"Erdos184.erdos_184.variants.conlon_fox_sudakov":{"url":"/FormalConjectures/ErdosProblems/«184»/#Erdos184___erdos_184___variants___conlon_fox_sudakov","anchor":"Erdos184___erdos_184___variants___conlon_fox_sudakov","docHtml":"\n Conlon, Fox, and Sudakov [CFS14] proved that $O_\\epsilon(n)$ cycles and edges suffice if $G$ has\nminimum degree at least $\\epsilon n$, for any $\\epsilon>0$.
"},"Erdos458.lcm_upto":{"url":"/FormalConjectures/ErdosProblems/«458»/#Erdos458___lcm_upto","anchor":"Erdos458___lcm_upto","docHtml":"\n The least common multiple of the integers in the set ${1, \\dots, n}$.
"},"Erdos458.erdos_458":{"url":"/FormalConjectures/ErdosProblems/«458»/#Erdos458___erdos_458","anchor":"Erdos458___erdos_458","docHtml":"\n Let $\\operatorname{lcm}(1, \\dots, n)$ denote the least common multiple of ${1, \\dots, n}$.\nLet $p_k$ be the $k$-th prime.\nIs it true that for all $k \\geq 1$, $\\operatorname{lcm}(1, \\dots, p_{k+1}-1) < p_k \\cdot \\operatorname{lcm}(1, \\dots, p_k)$?
"},"Erdos233.erdos_233":{"url":"/FormalConjectures/ErdosProblems/«233»/#Erdos233___erdos_233","anchor":"Erdos233___erdos_233","docHtml":"\n A conjecture by Heath-Brown:\nThe sum of squares of the first $N$ gaps between consecutive primes behaves like $N * (log N)^2$.
"},"Erdos233.erdos_233.variants.upper_bound":{"url":"/FormalConjectures/ErdosProblems/«233»/#Erdos233___erdos_233___variants___upper_bound","anchor":"Erdos233___erdos_233___variants___upper_bound","docHtml":"\n Cramér proved an upper bound of $O(N(\\log N)^4)$ conditional on the Riemann hypothesis.
"},"Erdos233.erdos_233.variants.lower_bound":{"url":"/FormalConjectures/ErdosProblems/«233»/#Erdos233___erdos_233___variants___lower_bound","anchor":"Erdos233___erdos_233___variants___lower_bound","docHtml":"\n The prime number theorem immediately implies a lower bound of $\\gg N(\\log N)^2$ for the sum of\nsquares of gaps between consecutive primes.
\n\n Formal proof linked here provided by AlphaProof.
"},"Erdos25.erdos_25":{"url":"/FormalConjectures/ErdosProblems/«25»/#Erdos25___erdos_25","anchor":"Erdos25___erdos_25","docHtml":"\n Let $n_1 < n_2 < \\dots$ be an arbitrary sequence of integers, each with an associated residue class\n$a_i \\pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n < n_i$ or\n$n \\not\\equiv a_i \\pmod{n_i}$. Must the logarithmic density of $A$ exist?
"},"Erdos123.IsDComplete":{"url":"/FormalConjectures/ErdosProblems/«123»/#Erdos123___IsDComplete","anchor":"Erdos123___IsDComplete","docHtml":"\n A set A of natural numbers is d-complete if every sufficiently large integer\nis the sum of distinct elements of A such that no element divides another.
\n Reference: [ErLe96] Erdős, P. and Lewin, M.,
\n Characterizes a \"snug\" finite set of natural numbers:\nall elements are within a multiplicative factor $(1 + ε)$ of the minimum.\nSpecifically, for a finite set $A$ and $ε > 0$, all $a ∈ A$ satisfy $a < (1 + ε) · min(A)$.
"},"Erdos123.PairwiseCoprime":{"url":"/FormalConjectures/ErdosProblems/«123»/#Erdos123___PairwiseCoprime","anchor":"Erdos123___PairwiseCoprime","docHtml":"\n Predicate for pairwise coprimality of three integers.\nRequires all three input values to be pairwise coprime to each other.
"},"Erdos123.erdos_123":{"url":"/FormalConjectures/ErdosProblems/«123»/#Erdos123___erdos_123","anchor":"Erdos123___erdos_123","docHtml":"\nErdős Problem #123
\n\n Let $a, b, c$ be three integers which are pairwise coprime. Is every large integer\nthe sum of distinct integers of the form $a^k b^l c^m$ ($k, l, m ≥ 0$), none of which\ndivide any other?
\n\n Equivalently: is the set ${a^k b^l c^m : k, l, m \\geq 0}$ d-complete?
\n\n Note: For this not to reduce to the two-integer case, we need the integers\nto be greater than one and distinct.
"},"Erdos123.erdos_123.variants.erdos_lewin_3_5_7":{"url":"/FormalConjectures/ErdosProblems/«123»/#Erdos123___erdos_123___variants___erdos_lewin_3_5_7","anchor":"Erdos123___erdos_123___variants___erdos_lewin_3_5_7","docHtml":"\n Erdős and Lewin proved this conjecture when $a = 3$, $b = 5$, and $c = 7$.
\n\n Reference: [ErLe96] Erdős, P. and Lewin, Mordechai,\n
\n A simpler case: the set of numbers of the form $2^k 3^l$ ($k, l ≥ 0$) is d-complete.
\n\n This was initially conjectured by Erdős in 1992, who called it a \"nice and difficult\"\nproblem, but it was quickly proven by Jansen and others using a simple inductive argument:
\n\n If $n = 2m$ is even, apply the inductive hypothesis to $m$ and double all summands.
\n\n If $n$ is odd, let $3^k$ be the largest power of $3$ with $3^k ≤ n$, and apply the\ninductive hypothesis to $n - 3^k$ (which is even).
\n\n Reference: [Er92b] Erdős, Paul,
\n A stronger conjecture for numbers of the form $2^k 3^l 5^j$.
\n\n For any $ε > 0$, all large integers $n$ can be written as the sum of distinct integers\n$b_1 < ... < b_t$ of the form $2^k 3^l 5^j$ where $b_t < (1 + ϵ) b_1$.
"},"Erdos28.erdos_28":{"url":"/FormalConjectures/ErdosProblems/«28»/#Erdos28___erdos_28","anchor":"Erdos28___erdos_28","docHtml":"\n If $A ⊆ \\mathbb{N}$ is such that $A + A$ contains all but finitely many integers then\n$\\limsup 1_A ∗ 1_A(n) = \\infty$.
"},"Erdos1102.HasPropertyP":{"url":"/FormalConjectures/ErdosProblems/«1102»/#Erdos1102___HasPropertyP","anchor":"Erdos1102___HasPropertyP","docHtml":"\n Property P : A set $A ⊆ ℕ $ has property P, if for all $n ≥ 1$ the set\n$ {a ∈ A | n + a\\text{ is squarefree}}$ is finite.
"},"Erdos1102.HasPropertyQ":{"url":"/FormalConjectures/ErdosProblems/«1102»/#Erdos1102___HasPropertyQ","anchor":"Erdos1102___HasPropertyQ","docHtml":"\n Property Q : A set $A ⊆ ℕ $ has property Q, if the set\n${n ∈ ℕ | ∀ a ∈ A, n > a\\text{ implies }n + a\\text{ is squarefree}}$ is infinite.
"},"Erdos1102.erdos_1102.density_zero_of_P":{"url":"/FormalConjectures/ErdosProblems/«1102»/#Erdos1102___erdos_1102___density_zero_of_P","anchor":"Erdos1102___erdos_1102___density_zero_of_P","docHtml":"A = {a₁ < a₂ < …} has property P,\nthen A has natural density 0.\nEquivalently, (a_j / j) → ∞ as j → ∞.f : ℕ → ℕ that goes to infinity,\nthere exists a strictly increasing sequence A = {a₁ < a₂ < …}\nwith property P such that (a_j / j) ≤ f(j) for all j.6 / π^2.A has property Q and natural density 6 / π^2.\nEquivalently, (j / a_j) → 6/π^2 as j → ∞.\n
\n Let $h_1(n) = h(n)$ and $h_k(n) = h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist\n$i$ and $j$ such that $h_i(m) = h_j(n)$?
"},"Erdos383.erdos_383":{"url":"/FormalConjectures/ErdosProblems/«383»/#Erdos383___erdos_383","anchor":"Erdos383___erdos_383","docHtml":"\n Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime\ndivisor of\n$$\n\\prod_{i = 0}^k (p ^ 2 + i)\n$$\nis $p$?
"},"Erdos266.erdos_266":{"url":"/FormalConjectures/ErdosProblems/«266»/#Erdos266___erdos_266","anchor":"Erdos266___erdos_266","docHtml":"\n Let $a_n$ be an infinite sequence of positive integers such that $\\sum \\frac{1}{a_n}$ converges.\nThere exists some integer $t \\ge 1$ such that $\\sum \\frac{1}{a_n + t}$ is irrational.
\n\n This was disproven by Kovač and Tao in [KoTa24].
\n\n [KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series.\narXiv:2406.17593 (2024).
"},"Erdos266.erdos_266.variants.all_rationals":{"url":"/FormalConjectures/ErdosProblems/«266»/#Erdos266___erdos_266___variants___all_rationals","anchor":"Erdos266___erdos_266___variants___all_rationals","docHtml":"\n In fact, Kovač and Tao proved in [KoTa24] that there exists a strictly increasing\nsequence $a_n$ of positive integers such that $\\sum \\frac{1}{a_n + t}$ converges to a rational\nnumber for all $t \\in \\mathbb{Q}$ such that $t \\ne -a_n$ for any $n$.
\n\n [KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series.\narXiv:2406.17593 (2024).
"},"Erdos598.κ":{"url":"/FormalConjectures/ErdosProblems/«598»/#Erdos598______","anchor":"Erdos598______","docHtml":"\n Let $\\kappa = (2^{\\aleph_0})^+$. This is the successor cardinal of the continuum.
"},"Erdos598.erdos_598":{"url":"/FormalConjectures/ErdosProblems/«598»/#Erdos598___erdos_598","anchor":"Erdos598___erdos_598","docHtml":"\nErdős Problem 598:\nLet $m$ be an infinite cardinal and $\\kappa$ be the successor cardinal of $2^{\\aleph_0}$.\nCan one colour the countable subsets of $m$ using $\\kappa$ many colours so that every\n$X \\subseteq m$ with $|X| = \\kappa$ contains subsets of all possible colours?
"},"Erdos160.erdos_160.h":{"url":"/FormalConjectures/ErdosProblems/«160»/#Erdos160___erdos_160___h","anchor":"Erdos160___erdos_160___h","docHtml":"\n Let $h(n)$ be the smallest $k$ such that ${1,\\ldots,n}$ can be coloured with $k$ colours\nso that every four-term arithmetic progression must contain at least three distinct colours.
"},"Erdos160.erdos_160.known_upper":{"url":"/FormalConjectures/ErdosProblems/«160»/#Erdos160___erdos_160___known_upper","anchor":"Erdos160___erdos_160___known_upper","docHtml":"\n On Mathoverflow user\nleechlattice shows that\n$h(n) \\ll n^{\\frac 2 3}$.
"},"Erdos160.erdos_160.better_upper":{"url":"/FormalConjectures/ErdosProblems/«160»/#Erdos160___erdos_160___better_upper","anchor":"Erdos160___erdos_160___better_upper","docHtml":"\n Estimate $h(n)$ by finding a better upper bound.
"},"Erdos160.erdos_160.better_lower":{"url":"/FormalConjectures/ErdosProblems/«160»/#Erdos160___erdos_160___better_lower","anchor":"Erdos160___erdos_160___better_lower","docHtml":"\n Estimate $h(n)$ by finding a better lower bound.
"},"Erdos160.erdos_160.variants.known_lower":{"url":"/FormalConjectures/ErdosProblems/«160»/#Erdos160___erdos_160___variants___known_lower","anchor":"Erdos160___erdos_160___variants___known_lower","docHtml":"\n The observation of Zachary Hunter in that question\ncoupled with the bounds of Kelley-Meka KeMe23 imply that\n$$h(N) \\gg \\exp(c(\\log N)^{\\frac 1 {12}})$$\nfor some $c > 0$.
"},"Erdos655.IsValid":{"url":"/FormalConjectures/ErdosProblems/«655»/#Erdos655___IsValid","anchor":"Erdos655___IsValid","docHtml":"\n A collection $x_1, \\dots, x_n\\in\\mathbb{R}^2$ is
\n Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that no circle whose centre is one\nof the $x_i$ contains three other points. Are there at least\n$$(1+c)\\frac{n}{2}$$\ndistinct distances determined between the $x_i$, for some constant $c>0$ and\nall $n$ sufficiently large?
\n\n Zach Hunter has observed that taking $n$ points equally spaced on a circle\ndisproves one natural interpretation of this conjecture.
"},"Erdos655.erdos_655.variants.general_position":{"url":"/FormalConjectures/ErdosProblems/«655»/#Erdos655___erdos_655___variants___general_position","anchor":"Erdos655___erdos_655___variants___general_position","docHtml":"\n Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that no circle whose centre is one\nof the $x_i$ contains three other points. Are there at least$$(1+c)\\frac{n}{2}$$\ndistinct distances determined between the $x_i$, for some constant $c>0$ and\nall $n$ sufficiently large?
\n\n In the spirit of related conjectures of Erdős and others, presumably\nsome kind of assumption that the points are in general position\n(e.g. no three on a line and no four on a circle) was intended.
"},"Erdos1145.Erdos1145Prop":{"url":"/FormalConjectures/ErdosProblems/«1145»/#Erdos1145___Erdos1145Prop","anchor":"Erdos1145___Erdos1145Prop","docHtml":"\n Let $A={1\\leq a_1 < a_2 < \\cdots}$ and $B={1\\leq b_1 < b_2 < \\cdots}$ be sets of integers with\n$a_n/b_n\\to 1$.
\n\n If $A+B$ contains all sufficiently large positive integers then is it true that\n$\\limsup 1_A\\ast 1_B(n)=\\infty$?
\n\n Formalization note: There's some discussion in the comments of [erdosproblems.com/28] and\n[erdosproblems.com/1145] about whether or not $0$ should be included in $A$ or $B$ and has been\nleft purposely ambiguous. Problem 1145 was originally written as $A + B = \\mathbb{N}$, which\nwould imply that $0$ would need to exist in $A$ or $B$ to include $1$ in $A + B$. However, it's been\nmade more general and rewritten as \"sufficiently large positive integers\". The formalization below\nis the version that includes $0$.
"},"Erdos1145.erdos_1145":{"url":"/FormalConjectures/ErdosProblems/«1145»/#Erdos1145___erdos_1145","anchor":"Erdos1145___erdos_1145","docHtml":"\n Let $A={1\\leq a_1 < a_2 < \\cdots}$ and $B={1\\leq b_1 < b_2 < \\cdots}$ be sets of integers with\n$a_n/b_n\\to 1$.
\n\n If $A+B$ contains all sufficiently large positive integers then is it true that\n$\\limsup 1_A\\ast 1_B(n)=\\infty$?
\n\n A conjecture of Erdős and Sárközy.
"},"Erdos1145.erdos_1145.test_implies_erdos_28":{"url":"/FormalConjectures/ErdosProblems/«1145»/#Erdos1145___erdos_1145___test_implies_erdos_28","anchor":"Erdos1145___erdos_1145___test_implies_erdos_28","docHtml":"\n A stronger form of [erdosproblems.com/28].
"},"Erdos348.erdos_348":{"url":"/FormalConjectures/ErdosProblems/«348»/#Erdos348___erdos_348","anchor":"Erdos348___erdos_348","docHtml":"\n For what values of $0 \\leq m < n$ is there a complete sequence\n$A = {a_1 \\leq a_2 \\leq \\cdots}$ of integers such that
\n\n $A$ remains complete after removing any $m$ elements, but
\n\n $A$ is not complete after removing any $n$ elements.
\n\n Are there infinitely many integers not of the form $n - \\phi(n)$?
\n\n Asked by Erdős and Sierpiński. Numbers not of the form we call non-cototients.
\n\n Browkin and Schinzel [BrSc95] provided an affirmative answer to this question, proving that any\ninteger of the shape $2^{k}\\cdot 509203$ for $k\\geq 1$ is a non-cototient.
\n\n This is discussed in problem B36 of Guy's collection [Gu04].
\n\n This was formalized in Lean by Alexeev using Aristotle.
"},"Erdos418.erdos_418.variants.conditional":{"url":"/FormalConjectures/ErdosProblems/«418»/#Erdos418___erdos_418___variants___conditional","anchor":"Erdos418___erdos_418___variants___conditional","docHtml":"\n It follows from a slight strengthening of the Goldbach conjecture that every odd number can be\nwritten as $n - \\phi(n)$.\nIn particular, we assume that every even number greater than 6 can be written as the sum of two\n
\n Erdős [Er73b] has shown that a positive density set of natural numbers cannot be written as\n$\\sigma(n)-n$ (numbers not of this form are called nonaliquot, or sometimes untouchable).
"},"Erdos418.erdos_418.variants.soln":{"url":"/FormalConjectures/ErdosProblems/«418»/#Erdos418___erdos_418___variants___soln","anchor":"Erdos418___erdos_418___variants___soln","docHtml":"\n A solution to erdos_418 was shown by Browkin and Schinzel [BrSc95] by showing that any integer of\nthe form $2^(k + 1)\\cdot 509203$ is not of the form $n - \\phi(n)$.
"},"Erdos418.erdos_418.variants.density":{"url":"/FormalConjectures/ErdosProblems/«418»/#Erdos418___erdos_418___variants___density","anchor":"Erdos418___erdos_418___variants___density","docHtml":"\n It is open whether the set of non-cototients has positive density.
"},"Erdos257.erdos_257":{"url":"/FormalConjectures/ErdosProblems/«257»/#Erdos257___erdos_257","anchor":"Erdos257___erdos_257","docHtml":"\n Let $A\\subseteq\\mathbb{N}$ be an infinite set. Is\n$$\n\\sum_{n\\in A} \\frac{1}{2^n - 1}\n$$\nirrational?
"},"Erdos257.erdos_257.variants.tsum_top_eq":{"url":"/FormalConjectures/ErdosProblems/«257»/#Erdos257___erdos_257___variants___tsum_top_eq","anchor":"Erdos257___erdos_257___variants___tsum_top_eq","docHtml":"\n Show that\n$$\n\\sum_{n} \\frac{1}{2^n - 1} = \\sum_{n} \\frac{d(n)}{2^n},\n$$\nwhere $d(n)$ is the number of divisors of $n$.
"},"Erdos257.erdos_257.variants.tsum_top":{"url":"/FormalConjectures/ErdosProblems/«257»/#Erdos257___erdos_257___variants___tsum_top","anchor":"Erdos257___erdos_257___variants___tsum_top","docHtml":"\n Show that\n$$\n\\sum_{n} \\frac{d(n)}{2^n}\n$$\nis irrational.
\n\n [Er48] Erdős, P.,
\n Is\n$$\n\\sum_{n\\geq 2}\\frac{\\omega(n)}{2^n}\n$$\nirrational? (Here $\\omega(n)$ counts the number of distinct prime divisors of $n$.)
"},"Erdos69.erdos_69.variants.specialisation_of_erdos_257":{"url":"/FormalConjectures/ErdosProblems/«69»/#Erdos69___erdos_69___variants___specialisation_of_erdos_257","anchor":"Erdos69___erdos_69___variants___specialisation_of_erdos_257","docHtml":"\n Tao observed that erdos_69 is a special case of erdos_257, since\n$$\n\\sum_{n\\geq 2}\\frac{\\omega(n)}{2^n} = \\sum_p \\frac{1}{2^p - 1}.\n$$
\n Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$\nvertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform\nhypergraph on $n$ vertices.
\n\n Is there some constant $c>0$ such that\n$$ R_3(n) \\geq 2^{2^{cn}}? $$
"},"Erdos212.erdos_212":{"url":"/FormalConjectures/ErdosProblems/«212»/#Erdos212___erdos_212","anchor":"Erdos212___erdos_212","docHtml":"\n Is there a dense subset of ℝ^2 such that all pairwise distances\nare rational?
"},"Erdos373.S":{"url":"/FormalConjectures/ErdosProblems/«373»/#Erdos373___S","anchor":"Erdos373___S","docHtml":"\n Let S be the set of non-trivial solutions to the equation n! = a₁! ··· aₖ!\nsuch that a₁ ≥ ... ≥ aₖ and n-1 > a₁.
\n Show that the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has\nonly finitely many solutions.
\n Show that if P(n(n+1)) / log n → ∞ where P(m) denotes the largest prime factor of m, then\nthe equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only\nfinitely many solutions.
\n Show that if P(n(n−1)) > 4 log n for large enough n, where P(m) denotes the\nlargest prime factor of m, then the equation n!=a_1!a_2!···a_k!, with\nn−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions.
\n Hickerson conjectured the largest solution the equation n!=a_1!a_2!···a_k!, with\nn−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, is 16!=14!5!2!.
\n Surányi was the first to conjecture that the only non-trivial solution to a!b!=n!\nis 6!7!=10!.
\n The maximum size of a Sidon set in {1, ..., N} is less than or equal to 2 * √N.
\nErdős Problem 44: Let N ≥ 1 and A ⊆ {1,…,N} be a Sidon set. Is it true that, for any ε > 0,\nthere exist M = M(ε) and B ⊆ {N+1,…,M} such that A ∪ B ⊆ {1,…,M} is a Sidon set\nof size at least (1−ε)M^{1/2}?
\n This problem asks whether any Sidon set can be extended to achieve a density\narbitrarily close to the optimal density for Sidon sets.
"},"Erdos44.erdos_44.variants.empty_start":{"url":"/FormalConjectures/ErdosProblems/«44»/#Erdos44___erdos_44___variants___empty_start","anchor":"Erdos44___erdos_44___variants___empty_start","docHtml":"\n The case where we start with an empty set (constructing large Sidon sets).
"},"Erdos44.example_sidon_set":{"url":"/FormalConjectures/ErdosProblems/«44»/#Erdos44___example_sidon_set","anchor":"Erdos44___example_sidon_set","docHtml":"\n The set {1, 2, 4, 8, 13} is a Sidon set in {1, ..., 13}.
\n For any N, there exists a Sidon set of size at least √N/2.
\n The greedy construction gives a Sidon set of size approximately √N.
\n
\n Let $f(n)$ be maximal such that if $A\\subseteq\\mathbb{N}$ has $|A| = n$ then\n$\\prod_{a\\neq b\\in A}(a + b)$ has at least $f(n)$ distinct prime factors.\nIs it true that $\\frac{f(n)}{\\log n} \\to\\infty$?
"},"Erdos126.erdos_126.variants.IsBigO":{"url":"/FormalConjectures/ErdosProblems/«126»/#Erdos126___erdos_126___variants___IsBigO","anchor":"Erdos126___erdos_126___variants___IsBigO","docHtml":"\n Erdős and Turán proved [ErTu34] in their first joint paper that\n$$\n\\log n \\ll f(n) \\ll \\frac{n}{\\log n}\n$$
\n\n [ErTu34] Erdős, Paul and Turan, Paul,
\n Erdős says that $f(n) = o(\\frac{n}{\\log n})$ has never been proved.
"},"Erdos978.erdos_978.variants.sub_one":{"url":"/FormalConjectures/ErdosProblems/«978»/#Erdos978___erdos_978___variants___sub_one","anchor":"Erdos978___erdos_978___variants___sub_one","docHtml":"\n Let f ∈ ℤ[X] be an irreducible polynomial with positive leading coefficient. Suppose that the\ndegree k of f is larger than 2 and is not equal to a power of 2. Then the set of n such\nthat f n is (k - 1)-th power free is infinite, and this is proved in [Er53].
\n Let f ∈ ℤ[X] be an irreducible polynomial with positive leading coefficient. Suppose that the\ndegree k of f is larger than 2, is not equal to a power of 2, and f n has no fixed\n(k - 1)-th power divisors other than 1. Then the set of n such that f n is (k - 1)-th\npower free has positive density, and this is proved in [Ho67].
\n If the degree k of f is larger than or equal to 9, then the set of n such that f n is\n(k - 2)-th power free has infinitely many elements. This result is proved in [Br11].
\n If $k > 3$ (and $k \\neq 2^l$), then are there infinitely many $n$ for which $f(n)$ is\n$(k-2)$-power-free?
\n\n This was disproved by the DeepMind prover agent.
"},"Erdos978.erdos_978.parts.ii":{"url":"/FormalConjectures/ErdosProblems/«978»/#Erdos978___erdos_978___parts___ii","anchor":"Erdos978___erdos_978___parts___ii","docHtml":"\n If $k>3$ (and $k \\neq 2^l$), and for all primes $p$ there exists $n$ such that $p^{k-2}\\nmid f(n)$,\nthen are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free?
"},"Erdos978.erdos_978.parts.iii":{"url":"/FormalConjectures/ErdosProblems/«978»/#Erdos978___erdos_978___parts___iii","anchor":"Erdos978___erdos_978___parts___iii","docHtml":"\n Does n ^ 4 + 2 represent infinitely many squarefree numbers?
\n Is it true that, for every $c$, there exists an $n$ such that $\\sigma(n)>cn$ but there is no\ncovering system whose moduli all divide $n$?
\n\n This was answered affirmatively by Haight [Ha79].
"},"SteinerSystems.SteinerSystem":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___SteinerSystem","anchor":"SteinerSystems___SteinerSystem","docHtml":"\n An $S(t, k, n)$-Steiner system is a collection of $k$-element subsets (called blocks) of\n${0, \\ldots, n-1}$ such that every $t$-element subset is contained in exactly one block.
\n\n This is the standard notation from combinatorics, where:
\n\n $n$ is the number of points
\n\n $k$ is the block size
\n\n $t$ is the covering parameter (every $t$-subset is in exactly one block)
\n\n The blocks of the Steiner system.
"},"SteinerSystems.SteinerSystem.block_card":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___SteinerSystem___block_card","anchor":"SteinerSystems___SteinerSystem___block_card","docHtml":"\n Every block has exactly $k$ elements.
"},"SteinerSystems.SteinerSystem.cover_unique":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___SteinerSystem___cover_unique","anchor":"SteinerSystems___SteinerSystem___cover_unique","docHtml":"\n Every $t$-element subset is contained in exactly one block.\nNote: We use .filter.card = 1 instead of ∃! because ∃! B ∈ blocks, R ⊆ B desugars to\na quantifier over all Finset (Fin n), which loses Decidable and breaks native_decide.
\n A constructive witness for a large Steiner system: concrete values of $n$, $k$, $t$\nsatisfying $n > k > t > 5$, $t < 10$, $n < 200$, together with an explicit Steiner system.
"},"SteinerSystems.LargeSteinerSystemWitness.n":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___LargeSteinerSystemWitness___n","anchor":"SteinerSystems___LargeSteinerSystemWitness___n","docHtml":"\n The size of the ground set.
"},"SteinerSystems.LargeSteinerSystemWitness.k":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___LargeSteinerSystemWitness___k","anchor":"SteinerSystems___LargeSteinerSystemWitness___k","docHtml":"\n The block size.
"},"SteinerSystems.LargeSteinerSystemWitness.t":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___LargeSteinerSystemWitness___t","anchor":"SteinerSystems___LargeSteinerSystemWitness___t","docHtml":"\n The covering parameter.
"},"SteinerSystems.LargeSteinerSystemWitness.h_nk":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___LargeSteinerSystemWitness___h_nk","anchor":"SteinerSystems___LargeSteinerSystemWitness___h_nk","docHtml":"\n The covering parameter.
"},"SteinerSystems.LargeSteinerSystemWitness.h_kt":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___LargeSteinerSystemWitness___h_kt","anchor":"SteinerSystems___LargeSteinerSystemWitness___h_kt","docHtml":"\n The covering parameter.
"},"SteinerSystems.LargeSteinerSystemWitness.h_t_lower":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___LargeSteinerSystemWitness___h_t_lower","anchor":"SteinerSystems___LargeSteinerSystemWitness___h_t_lower","docHtml":"\n The covering parameter.
"},"SteinerSystems.LargeSteinerSystemWitness.h_t_upper":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___LargeSteinerSystemWitness___h_t_upper","anchor":"SteinerSystems___LargeSteinerSystemWitness___h_t_upper","docHtml":"\n The covering parameter.
"},"SteinerSystems.LargeSteinerSystemWitness.h_n_upper":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___LargeSteinerSystemWitness___h_n_upper","anchor":"SteinerSystems___LargeSteinerSystemWitness___h_n_upper","docHtml":"\n The covering parameter.
"},"SteinerSystems.LargeSteinerSystemWitness.system":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___LargeSteinerSystemWitness___system","anchor":"SteinerSystems___LargeSteinerSystemWitness___system","docHtml":"\n The explicit Steiner system.
"},"SteinerSystems.large_steiner_systems":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___large_steiner_systems","anchor":"SteinerSystems___large_steiner_systems","docHtml":"\n Construct an $S(t, k, n)$-Steiner system with $n > k > t > 5$, $t < 10$, and $n < 200$.
\n\n No example of a Steiner system with $t > 5$ is known, despite a 2014 existence theorem\nby Keevash showing that such systems must exist for sufficiently large $n$.
\n\n
\n Sanity check: the Fano plane is an $S(2, 3, 7)$-Steiner system.
\n\n The Fano plane consists of $7$ blocks of size $3$ over $7$ points,\nwhere every pair of points is contained in exactly one block.
"},"SteinerSystems.steiner_system_5_6_12":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___steiner_system_5_6_12","anchor":"SteinerSystems___steiner_system_5_6_12","docHtml":"\nExistence of $S(5, 6, 12)$: The small Witt design.
\n\n There exists a unique Steiner system $S(5, 6, 12)$, known as the small Witt design.\nIt was constructed by Witt (1938) and is closely related to the Mathieu group $M_{12}$.\nThis is one of only two known Steiner systems with $t = 5$.
"},"SteinerSystems.steiner_system_5_8_24":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___steiner_system_5_8_24","anchor":"SteinerSystems___steiner_system_5_8_24","docHtml":"\nExistence of $S(5, 8, 24)$: The large Witt design.
\n\n There exists a unique Steiner system $S(5, 8, 24)$, known as the large Witt design.\nIt was constructed by Witt (1938) and is closely related to the Mathieu group $M_{24}$.\nThis is one of only two known Steiner systems with $t = 5$.
"},"SteinerSystems.infinitely_many_steiner_t4":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___infinitely_many_steiner_t4","anchor":"SteinerSystems___infinitely_many_steiner_t4","docHtml":"\nThere are infinitely many Steiner systems with $t = 4$.
\n\n Keevash (2014) proved that for any fixed $t$ and $k$, a Steiner system $S(t, k, n)$\nexists for all sufficiently large $n$ satisfying the necessary divisibility conditions.\nSince there are infinitely many such admissible $n$, this implies infinitely many\n$S(4, k, n)$ systems exist (for any fixed $k > 4$). The proof is nonconstructive.
\n\n Explicit examples include $S(4, 5, 11)$ (the unique system, related to the Mathieu\ngroup $M_{11}$) and $S(4, 7, 23)$ (related to the Mathieu group $M_{23}$).
"},"SteinerSystems.infinitely_many_steiner_t5":{"url":"/FormalConjectures/Wikipedia/SteinerSystem/#SteinerSystems___infinitely_many_steiner_t5","anchor":"SteinerSystems___infinitely_many_steiner_t5","docHtml":"\nThere are infinitely many Steiner systems with $t = 5$.
\n\n Keevash (2014) proved that for any fixed $t$ and $k$, a Steiner system $S(t, k, n)$\nexists for all sufficiently large $n$ satisfying the necessary divisibility conditions.\nThis settles the long-standing open problem of whether infinitely many $S(5, k, n)$\nsystems exist. The proof is nonconstructive.
\n\n Only two explicit examples are known: $S(5, 6, 12)$ and $S(5, 8, 24)$, both Witt\ndesigns related to the Mathieu groups $M_{12}$ and $M_{24}$ respectively.\nNo Steiner system with $t \\geq 6$ has been explicitly constructed, though Keevash's\nresult guarantees their existence nonconstructively as well.
"},"CernyConjecture.cerny_conjecture":{"url":"/FormalConjectures/Wikipedia/CernyConjecture/#CernyConjecture___cerny_conjecture","anchor":"CernyConjecture___cerny_conjecture","docHtml":"\nČerný Conjecture: Every synchronizing DFA with $n$ states admits a\nsynchronizing word of length at most $(n - 1)^2$.
"},"CernyConjecture.shitov_upper_bound":{"url":"/FormalConjectures/Wikipedia/CernyConjecture/#CernyConjecture___shitov_upper_bound","anchor":"CernyConjecture___shitov_upper_bound","docHtml":"\nShitov's bound (2019): Every synchronizing DFA with $n$ states admits a synchronizing\nword of length at most $\\left(\\frac{7}{48} + \\frac{2 \\cdot 15625}{1597536}\\right) n^3 + o(n^3)$,\nwhere the $o(n^3)$ term is uniform over all alphabets. This is the best known upper bound\ntowards the Černý conjecture.
"},"FeitThompsonPrimeConjecture.feit_thompson_primes":{"url":"/FormalConjectures/Wikipedia/FeitThompsonPrimeConjecture/#FeitThompsonPrimeConjecture___feit_thompson_primes","anchor":"FeitThompsonPrimeConjecture___feit_thompson_primes","docHtml":"\n There are no distinct primes $p$ and $q$ such that $\\frac{q^p - 1}{q - 1}$ divides $\\frac{p^q - 1}{p - 1}$
"},"PierceBirkhoff.IsSemiAlgebraic":{"url":"/FormalConjectures/Wikipedia/PierceBirkhoff/#PierceBirkhoff___IsSemiAlgebraic","anchor":"PierceBirkhoff___IsSemiAlgebraic","docHtml":"\n A set is semi-algebraic in ℝⁿ if it can be described by a finite union of sets defined by\nmultivariate polynomial equations and inequalities.
\n A set is semi-algebraic in ℝ if it can be described by a finite boolean combination\nof polynomial equations and inequalities.
\n A function f : ℝⁿ → ℝ is piecewise polynomial if there exists a finite covering of ℝⁿ by\nclosed semi-algebraic sets such that the restriction of f to each set in the covering is\npolynomial.
\n A function f : ℝ → ℝ is piecewise polynomial if there exists a finite covering of ℝ by\nclosed semi-algebraic sets such that the restriction of f to each set in the covering is\npolynomial.
\n The Pierce-Birkhoff conjecture states that for every real piecewise-polynomial function\nf : ℝⁿ → ℝ, there exists a finite set of polynomials gᵢⱼ ∈ ℝ[x₁, ..., xₙ] such that\nf = supᵢ infⱼ(gᵢⱼ).
\n The Pierce-Birkhoff conjecture holds for n = 1.\nThis was proved by Louis Mahé.
\n The Pierce-Birkhoff conjecture holds for n = 2.\nThis was proved by Louis Mahé.
\nGilbreath's nth difference, $d^n$\nLet $d^0(n) = p_n$ and $d^k(n) = |d^{k-1}(n+1) - d^{k-1}(n)|
"},"Gilbreath.gilbreath_conjecture":{"url":"/FormalConjectures/Wikipedia/Gilbreath/#Gilbreath___gilbreath_conjecture","anchor":"Gilbreath___gilbreath_conjecture","docHtml":"\nGilbreath's conjecture\nGilbreath's conjecture states that every term in the sequence $d^k_0$ for $k > 0$ is equal to 1.
"},"QuadraticAlgebra.discr_rat_of_modEq_one":{"url":"/FormalConjectures/Wikipedia/WallSunSun/#QuadraticAlgebra___discr_rat_of_modEq_one","anchor":"QuadraticAlgebra___discr_rat_of_modEq_one","docHtml":"\n The discriminant of ℚ[√d] for d ≥ 2 squarefree congruent to 1 mod 4 is d.
\n The discriminant of ℚ[√d] for d ≥ 2 squarefree not congruent to 1 mod 4 is 4 * d.
\n A quadratic algebra L over a field K is isomorphic to the explicit quadratic algebra\nQuadraticAlgebra K a b for some a b : K.
\n An algebra L is quadratic over a field K iff it is isomorphic to the explicit quadratic\nalgebra QuadraticAlgebra K a b for some a b : K.
\n A quadratic number field K is isomorphic to the explicit quadratic field\nQuadraticAlgebra ℚ d 0 for some squarefree d : ℤ not equal to 1.
\n A number field K is quadratic iff it is isomorphic to the explicit quadratic field\nQuadraticAlgebra ℚ d 0 for some squarefree d : ℤ not equal to 1.
\n Fundamental discriminants are those integers D that appear as discriminants of quadratic\nfields.
\nD is a fundamental discriminant if it is either of the form 4m for m congruent to 2 or 3\nmod 4 squarefree, or if it congruent to 1 mod 4 and squarefree.
\n An integer D is a fundamental discriminant iff it is the discriminant of the explicit\nquadratic field QuadraticAlgebra ℚ d 0 for some squarefree d : ℤ not equal to 1.
\n An integer D is a fundamental discriminant iff it is the discriminant of some number field.
\n A prime $p$ is a Wall–Sun–Sun prime if and only if $L_p \\equiv 1 \\pmod{p^2}$, where $L_p$ is the\n$p$-th Lucas number. It is conjectured that there is at least one Wall–Sun–Sun prime.
"},"WallSunSun.infinite_isWallSunSunPrime":{"url":"/FormalConjectures/Wikipedia/WallSunSun/#WallSunSun___infinite_isWallSunSunPrime","anchor":"WallSunSun___infinite_isWallSunSunPrime","docHtml":"\n A prime $p$ is a Wall–Sun–Sun prime if and only if $L_p \\equiv 1 \\pmod{p^2}$, where $L_p$ is the\n$p$-th Lucas number. It is conjectured that there are infinitely many Wall-Sun-Sun primes.
"},"WallSunSun.infinite_isWallSunSunPrime_of_disc_eq":{"url":"/FormalConjectures/Wikipedia/WallSunSun/#WallSunSun___infinite_isWallSunSunPrime_of_disc_eq","anchor":"WallSunSun___infinite_isWallSunSunPrime_of_disc_eq","docHtml":"\n A Lucas–Wieferich prime associated with $(a,b)$ is an odd prime $p$, not dividing $a^2 - 4b$, such\nthat $U_{p-\\varepsilon}(a,b) \\equiv 0 \\pmod{p^2}$ where $U(a,b)$ is the Lucas sequence of the first\nkind and $\\varepsilon$ is the Legendre symbol $\\left({\\tfrac {a^2-4b}{p}}\\right)$.\nThe discriminant of this number is the quantity $a^2 - 4b$. It is conjectured that there are\ninfinitely many Lucas–Wieferich primes of any given non-one fundamental discriminant.
\n\n TODO: Source this conjecture
"},"NoetherProblem.IsRationalExtension":{"url":"/FormalConjectures/Wikipedia/NoetherProblem/#NoetherProblem___IsRationalExtension","anchor":"NoetherProblem___IsRationalExtension","docHtml":"\n A rational field extension is a field extension L/K isomorphic\nto a field of rational functions (in some arbitrary number of indeterminates.)
\n A rational field extension is a field extension L/K isomorphic\nto a field of rational functions (in some arbitrary number of indeterminates.)
\n If the index set ι is empty, then IsRationalExtension K L ι means that\nK, L are isomorphic as K algebras.
\n We say that a rational extension L of K has the H of the Galois group of L, the fixed field\nL^H is also a rational extension.
\n The Noether Problem: let L be the field of rational functions in n\nindeterminates over K. Is it true that L/K has the Noether property?
\n Solution: False.
"},"NoetherProblem.noether_problem.variants.two":{"url":"/FormalConjectures/Wikipedia/NoetherProblem/#NoetherProblem___noether_problem___variants___two","anchor":"NoetherProblem___noether_problem___variants___two","docHtml":"\n The Noether problem has a positive solution in the two indeterminate case.
"},"NoetherProblem.noether_problem.variants.three":{"url":"/FormalConjectures/Wikipedia/NoetherProblem/#NoetherProblem___noether_problem___variants___three","anchor":"NoetherProblem___noether_problem___variants___three","docHtml":"\n The Noether problem has a positive solution in the three indeterminate case.
"},"NoetherProblem.noether_problem.variants.four":{"url":"/FormalConjectures/Wikipedia/NoetherProblem/#NoetherProblem___noether_problem___variants___four","anchor":"NoetherProblem___noether_problem___variants___four","docHtml":"\n The Noether problem has a positive solution in the four indeterminate case.
"},"NoetherProblem.noether_problem.variants.forty_seven":{"url":"/FormalConjectures/Wikipedia/NoetherProblem/#NoetherProblem___noether_problem___variants___forty_seven","anchor":"NoetherProblem___noether_problem___variants___forty_seven","docHtml":"\n One can find a counterexample to the Noether Problem's claim by considering a\nrational function field in 47 indeterminates.
"},"JacobianConjecture.RegularFunction":{"url":"/FormalConjectures/Wikipedia/JacobianConjecture/#JacobianConjecture___RegularFunction","anchor":"JacobianConjecture___RegularFunction","docHtml":"\n Implicitly use σ as an index set and k as coefficient ring.
\n The Jacobian of a vector valued polynomial function, viewed as a polynomial.
"},"JacobianConjecture.RegularFunction.comp":{"url":"/FormalConjectures/Wikipedia/JacobianConjecture/#JacobianConjecture___RegularFunction___comp","anchor":"JacobianConjecture___RegularFunction___comp","docHtml":"\n The composition of two vector valued polynomial functions.
"},"JacobianConjecture.RegularFunction.id":{"url":"/FormalConjectures/Wikipedia/JacobianConjecture/#JacobianConjecture___RegularFunction___id","anchor":"JacobianConjecture___RegularFunction___id"},"JacobianConjecture.jacobian_conjecture":{"url":"/FormalConjectures/Wikipedia/JacobianConjecture/#JacobianConjecture___jacobian_conjecture","anchor":"JacobianConjecture___jacobian_conjecture","docHtml":"\n The Jacobian Conjecture: any regular function\n(i.e. vector valued polynomial function from) kⁿ → kᵐ\nwhose Jacobian is a non-zero constant has an inverse that\nis given by a regular function, where k is a field of characteristic 0
\n The evaluation of a regular function f over k at some point a\nwith coordinates in some algebra over k
\naeval is compatible with composition of regular functions.
\n The $n$-th tetrahedral number: $T_n = \\frac{n(n+1)(n+2)}{6}$.
"},"PollocksConjecture.NotSumOfFourTetrahedral":{"url":"/FormalConjectures/Wikipedia/PollocksConjecture/#PollocksConjecture___NotSumOfFourTetrahedral","anchor":"PollocksConjecture___NotSumOfFourTetrahedral","docHtml":"\n The set of natural numbers that are not a sum of $4$ tetrahedral numbers.
"},"PollocksConjecture.pollock_tetrahedral":{"url":"/FormalConjectures/Wikipedia/PollocksConjecture/#PollocksConjecture___pollock_tetrahedral","anchor":"PollocksConjecture___pollock_tetrahedral","docHtml":"\n Pollock's (tetrahedral numbers) conjecture:\nevery integer is the sum of at most $5$ tetrahedral numbers.
"},"PollocksConjecture.pollock_tetrahedral.salzer_levine":{"url":"/FormalConjectures/Wikipedia/PollocksConjecture/#PollocksConjecture___pollock_tetrahedral___salzer_levine","anchor":"PollocksConjecture___pollock_tetrahedral___salzer_levine","docHtml":"\n Salzer–Levine strengthening (as stated on Wikipedia/OEIS):\nthere are exactly $241$ integers that are not a sum of $4$ tetrahedral numbers, and the largest is $343867$.
"},"PollocksConjecture.pollock_tetrahedral.ncard_exceptions":{"url":"/FormalConjectures/Wikipedia/PollocksConjecture/#PollocksConjecture___pollock_tetrahedral___ncard_exceptions","anchor":"PollocksConjecture___pollock_tetrahedral___ncard_exceptions","docHtml":"\n As stated on Wikipedia/OEIS (A797), the set of exceptions has cardinality $241$.
"},"LehmerMahlerMeasureProblem.mahlerMeasure":{"url":"/FormalConjectures/Wikipedia/LehmerMahlerMeasureProblem/#LehmerMahlerMeasureProblem___mahlerMeasure","anchor":"LehmerMahlerMeasureProblem___mahlerMeasure","docHtml":"\n The Mahler measure of f(X) is defined as ‖a‖ ∏ᵢ max(1,‖αᵢ‖),\nwhere f(X)=a(X-α₁)(X-α₂)...(X-αₙ).
\n Let M(f) denote the Mahler measure of f.\nThere exists a constant μ>1 such that for any f(x)∈ℤ[x], M(f)>1 → M(f)≥μ.
\nμ=M(X^10 + X^9 - X^7 - X^6 - X^5 - X^4 - X^3 + X + 1) is the best value for lehmer_mahler_measure_problem.
\n If $f$ is not reciprocal and $M(f) > 1$ then $M(f) \\ge M(X^3 - X - 1)$.
"},"LehmerMahlerMeasureProblem.Polynomial.HasOddCoeffs":{"url":"/FormalConjectures/Wikipedia/LehmerMahlerMeasureProblem/#LehmerMahlerMeasureProblem___Polynomial___HasOddCoeffs","anchor":"LehmerMahlerMeasureProblem___Polynomial___HasOddCoeffs","docHtml":"\nPolynomial.HasOddCoeffs f means that all coefficients of f : Polynomial ℤ are odd.
\n If all the coefficients of $f$ are odd and $M(f) > 1$, then $M(f) \\ge M(X^2 - X - 1)$.
"},"Kaplansky.zero_divisor_conjecture":{"url":"/FormalConjectures/Wikipedia/Kaplansky/#Kaplansky___zero_divisor_conjecture","anchor":"Kaplansky___zero_divisor_conjecture","docHtml":"\nThe zero-divisor conjecture
\n\n If G is torsion-free, then the group algebra K[G] has no non-trivial zero divisors.
\nThe idempotent conjecture
\n\n If G is torsion-free, then K[G] has no non-trivial idempotents.
\n A unit in K[G] is trivial if it is exactly of the form kg where:
\nk is a unit in the base field K
\ng is an element of the group G
\nThe Promislow group ⟨ a, b | b⁻¹a²ba², a⁻¹b²ab² ⟩
\n The Promislow group is torsion-free.
"},"Kaplansky.UnitConjecture.counterexamples.i":{"url":"/FormalConjectures/Wikipedia/Kaplansky/#Kaplansky___UnitConjecture___counterexamples___i","anchor":"Kaplansky___UnitConjecture___counterexamples___i","docHtml":"\n If $P$ is the Promislow group, then the group ring $\\mathbb{F}_p[P]$ has a non-trivial unit.
"},"Kaplansky.UnitConjecture.counterexamples.ii":{"url":"/FormalConjectures/Wikipedia/Kaplansky/#Kaplansky___UnitConjecture___counterexamples___ii","anchor":"Kaplansky___UnitConjecture___counterexamples___ii","docHtml":"\n If $P$ is the Promislow group, then the group ring $\\mathbb{C}[P]$ has a non-trivial unit.
"},"Kaplansky.counter_unit_conjecture":{"url":"/FormalConjectures/Wikipedia/Kaplansky/#Kaplansky___counter_unit_conjecture","anchor":"Kaplansky___counter_unit_conjecture","docHtml":"\n The Unit Conjecture is false.
\n\n At least there is a counterexample for any prime and zero characteristic:\n[Mu21] Murray, A. (2021). More Counterexamples to the Unit Conjecture for Group Rings.\n[Pa21] Passman, D. (2021). On the counterexamples to the unit conjecture for group rings.\n[Ga24] Gardam, G. (2024). Non-trivial units of complex group rings.
"},"Kaplansky.counter_unit_conjecture_weak":{"url":"/FormalConjectures/Wikipedia/Kaplansky/#Kaplansky___counter_unit_conjecture_weak","anchor":"Kaplansky___counter_unit_conjecture_weak","docHtml":"\n There is a counterexample to Unit Conjecture in any characteristic.
"},"BabaiSeressConjectures.cayleyGraph":{"url":"/FormalConjectures/Wikipedia/DiameterSimpleFiniteGroups/#BabaiSeressConjectures___cayleyGraph","anchor":"BabaiSeressConjectures___cayleyGraph","docHtml":"\n The (undirected) Cayley graph of a group $G$ with respect to a generating set $S$.\nTwo elements $g, h \\in G$ are adjacent iff $g \\neq h$ and\n$g^{-1} h \\in S$ or $h^{-1} g \\in S$.
\n\n This is constructed using SimpleGraph.fromRel, which takes the relation\n$g \\sim h \\iff g^{-1} h \\in S$ and automatically symmetrizes it (via disjunction with the\nreverse relation) and enforces irreflexivity (via $g \\neq h$). In particular, this definition\neffectively uses the symmetrization $S \\cup S^{-1}$, so it produces a standard undirected\nCayley graph even when $S$ is not itself symmetric.
\n The diameter of a finite group $G$, defined as the maximum diameter of the Cayley graphs\n$\\Gamma(G, A)$ over all generating sets $A$ of $G$.
"},"BabaiSeressConjectures.groupDiam_fin_one":{"url":"/FormalConjectures/Wikipedia/DiameterSimpleFiniteGroups/#BabaiSeressConjectures___groupDiam_fin_one","anchor":"BabaiSeressConjectures___groupDiam_fin_one","docHtml":"\n For the trivial group (with one element), the group diameter is zero, since\nevery Cayley graph has only one vertex and hence diameter zero.
"},"BabaiSeressConjectures.groupDiam_alternating_three":{"url":"/FormalConjectures/Wikipedia/DiameterSimpleFiniteGroups/#BabaiSeressConjectures___groupDiam_alternating_three","anchor":"BabaiSeressConjectures___groupDiam_alternating_three","docHtml":"\n The alternating group $A_3 \\cong \\mathbb{Z}/3\\mathbb{Z}$ has group diameter $1$: every\nnon-trivial generating set produces a complete Cayley graph $K_3$, since any single non-identity\nelement and its inverse already reach the entire group.
"},"BabaiSeressConjectures.groupDiam_perm_two":{"url":"/FormalConjectures/Wikipedia/DiameterSimpleFiniteGroups/#BabaiSeressConjectures___groupDiam_perm_two","anchor":"BabaiSeressConjectures___groupDiam_perm_two","docHtml":"\n The symmetric group $S_2 \\cong \\mathbb{Z}/2\\mathbb{Z}$ has group diameter $1$: the unique\ngenerating set ${(01)}$ produces the complete graph $K_2$, since the single non-identity\nelement and its inverse (which are equal) cover the only other vertex.
"},"BabaiSeressConjectures.babai_seress_conjecture_alternating":{"url":"/FormalConjectures/Wikipedia/DiameterSimpleFiniteGroups/#BabaiSeressConjectures___babai_seress_conjecture_alternating","anchor":"BabaiSeressConjectures___babai_seress_conjecture_alternating","docHtml":"\nBabai–Seress Conjecture (Conjecture 1.5): There exists an absolute constant $C$ such\nthat the diameter of the alternating group $A_n$ satisfies\n$$\\operatorname{diam}(A_n) \\leq n^C.$$
\n"},"BabaiSeressConjectures.babai_seress_conjecture":{"url":"/FormalConjectures/Wikipedia/DiameterSimpleFiniteGroups/#BabaiSeressConjectures___babai_seress_conjecture","anchor":"BabaiSeressConjectures___babai_seress_conjecture","docHtml":"\nBabai–Seress Conjecture (Conjecture 1.7): There exists an absolute constant $C$ such\nthat every finite simple non-abelian group $G$ satisfies\n$$\\operatorname{diam}(G) \\leq (\\log |G|)^C.$$
\n"},"RamseyNumbers.IsGraphRamsey":{"url":"/FormalConjectures/Wikipedia/RamseyNumbers/#RamseyNumbers___IsGraphRamsey","anchor":"RamseyNumbers___IsGraphRamsey","docHtml":"\nIsGraphRamsey n k l means that for every simple graph G on n vertices, either
\nG contains a clique of size k, or
\n the complement graph Gᶜ contains a clique of size l (equivalently, G contains an\nindependent set of size l).
\n Monotonicity in the number of vertices.
"},"RamseyNumbers.IsGraphRamsey.symm":{"url":"/FormalConjectures/Wikipedia/RamseyNumbers/#RamseyNumbers___IsGraphRamsey___symm","anchor":"RamseyNumbers___IsGraphRamsey___symm","docHtml":"\n Symmetry in the clique / independent set sizes.
"},"RamseyNumbers.graphRamseyNumber":{"url":"/FormalConjectures/Wikipedia/RamseyNumbers/#RamseyNumbers___graphRamseyNumber","anchor":"RamseyNumbers___graphRamseyNumber","docHtml":"\n The (graph) Ramsey number R(k,l) is the least natural number n such that IsGraphRamsey n k l\nholds.
\n The open problem: determine the Ramsey number $R(5,5)$.
\n\n It is known that $43 \\le R(5,5) \\le 46$.
"},"RamseyNumbers.ramsey_number_five_five_lower_bound":{"url":"/FormalConjectures/Wikipedia/RamseyNumbers/#RamseyNumbers___ramsey_number_five_five_lower_bound","anchor":"RamseyNumbers___ramsey_number_five_five_lower_bound","docHtml":"\n Lower bound $43 \\le R(5,5)$, equivalently: there exists a graph on $42$ vertices with no\n$5$-clique and no independent set of size $5$.
"},"RamseyNumbers.ramsey_number_five_five_upper_bound":{"url":"/FormalConjectures/Wikipedia/RamseyNumbers/#RamseyNumbers___ramsey_number_five_five_upper_bound","anchor":"RamseyNumbers___ramsey_number_five_five_upper_bound","docHtml":"\n Upper bound $R(5,5) \\le 46$, i.e. every graph on $46$ vertices contains a $5$-clique or an\nindependent set of size $5$.
"},"InverseGalois.GaloisRealization":{"url":"/FormalConjectures/Wikipedia/InverseGalois/#InverseGalois___GaloisRealization","anchor":"InverseGalois___GaloisRealization","docHtml":"\n
\n
\n
\n
\n
\n
\n Say a group G is realizable over a field K if it\nis isomorphic to the Galois group of a Galois extension\nof K
\n Say a group G is realizable over a field K if it\nis isomorphic to the Galois group of a Galois extension\nof K
\n The Inverse Galois Problem: every finite group is\nisomorphic to the Galois group of a Galois extension of the\nrationals.
"},"InverseGalois.inverse_galois_problem.variants.cyclic":{"url":"/FormalConjectures/Wikipedia/InverseGalois/#InverseGalois___inverse_galois_problem___variants___cyclic","anchor":"InverseGalois___inverse_galois_problem___variants___cyclic","docHtml":"\n Every finite cyclic group is realizable.
"},"InverseGalois.inverse_galois_problem.variants.abelian":{"url":"/FormalConjectures/Wikipedia/InverseGalois/#InverseGalois___inverse_galois_problem___variants___abelian","anchor":"InverseGalois___inverse_galois_problem___variants___abelian","docHtml":"\n Every finite abelian group is realizable.
"},"InverseGalois.inverse_galois_problem.variants.symmetric_group":{"url":"/FormalConjectures/Wikipedia/InverseGalois/#InverseGalois___inverse_galois_problem___variants___symmetric_group","anchor":"InverseGalois___inverse_galois_problem___variants___symmetric_group","docHtml":"\n Every finite symmetric group is realizable.
"},"InverseGalois.inverse_galois_problem.variants.complex_rational_functions":{"url":"/FormalConjectures/Wikipedia/InverseGalois/#InverseGalois___inverse_galois_problem___variants___complex_rational_functions","anchor":"InverseGalois___inverse_galois_problem___variants___complex_rational_functions","docHtml":"\n Every finite group is realisable over the field of rational functions\nwith complex coefficients.
"},"InverseGalois.inverse_galois_problem.variants.complex_function_field":{"url":"/FormalConjectures/Wikipedia/InverseGalois/#InverseGalois___inverse_galois_problem___variants___complex_function_field","anchor":"InverseGalois___inverse_galois_problem___variants___complex_function_field","docHtml":"\n Every finite group is realisable over the field of rational functions\nwith coefficients K, where K is any field of characteristic 0.
\n The classic amicable pair $(220, 284)$.
"},"AmicableNumbers.IsAmicable.symm":{"url":"/FormalConjectures/Wikipedia/AmicableNumbers/#AmicableNumbers___IsAmicable___symm","anchor":"AmicableNumbers___IsAmicable___symm","docHtml":"\nIsAmicable is symmetric.
\nRelatively prime amicable numbers conjecture.\nDo there exist amicable numbers $(a, b)$ with $\\gcd(a, b) = 1$?
\n\n All known amicable pairs share a common factor. It is an open question\nwhether a pair of relatively prime amicable numbers can exist.
\n\n
\nInfinitely many amicable numbers conjecture.
\n\n Are there infinitely many pairs of amicable numbers?
\n\n While many amicable pairs are known, it remains open whether there are infinitely many.
\n\n
\nAmicable numbers with opposite parity conjecture.\nDo there exist amicable numbers $(a, b)$ where one is even and the other is odd?
\n\n All known amicable pairs are either both even or both odd. It is widely believed\nthat mixed-parity amicable pairs do not exist, but this remains open.
\n\n
\nAgrawal's Primality Conjecture.
\n\n Does the congruence $(X-1)^n \\equiv X^n - 1 \\pmod{n, X^r-1}$ imply\n$n$ is prime (with a specific exception for $n^2 \\equiv 1 \\pmod{r}$)?
\n\n While the \"if\" direction is a known theorem, the \"only if\" direction\nremains a conjecture.
"},"AgrawalConjecture.agrawal_conjecture.variants.popovych":{"url":"/FormalConjectures/Wikipedia/Agrawal/#AgrawalConjecture___agrawal_conjecture___variants___popovych","anchor":"AgrawalConjecture___agrawal_conjecture___variants___popovych","docHtml":"\nRoman B. Popovych Conjecture.\nA stronger version of Agrawal's conjecture, which also considers the congruence\n$(X+2)^n \\equiv X^n + 2 \\pmod{n, X^r-1}$.\nIf both congruences hold, then $n$ is either prime or $n^2 \\equiv 1 \\pmod{r}$.\nThis variant was proposed by Roman B. Popovych in 2018.
"},"LeinsterGroup.IsLeinster":{"url":"/FormalConjectures/Wikipedia/LeinsterGroup/#LeinsterGroup___IsLeinster","anchor":"LeinsterGroup___IsLeinster","docHtml":"\n A finite group G is a Leinster group if the sum of the orders of all its normal subgroups\nequals twice the group's order.
\nConjecture: Are there infinitely many Leinster groups?
\n\n This asks whether there exist infinitely many (non-isomorphic) finite groups that are\nLeinster groups.
\n\n Formalized via the negation of \"Does there exist an n such that all Leinster groups have\norder less than n\".
"},"LeinsterGroup.cyclic_of_perfect_is_leinster":{"url":"/FormalConjectures/Wikipedia/LeinsterGroup/#LeinsterGroup___cyclic_of_perfect_is_leinster","anchor":"LeinsterGroup___cyclic_of_perfect_is_leinster","docHtml":"\n Cyclic groups of perfect number order are Leinster groups.
\n\n This follows from the fact that for a cyclic group, all subgroups are normal and correspond\nto divisors of the group order, and a number is perfect if and only if the sum of its divisors\n(including itself) equals twice the number.
"},"LeinsterGroup.abelian_is_leinster_iff_cyclic_perfect":{"url":"/FormalConjectures/Wikipedia/LeinsterGroup/#LeinsterGroup___abelian_is_leinster_iff_cyclic_perfect","anchor":"LeinsterGroup___abelian_is_leinster_iff_cyclic_perfect","docHtml":"\n An abelian group is a Leinster group if and only if it is cyclic with order equal\nto a perfect number.
\n\n Reference: Leinster, Tom (2001). \"Perfect numbers and groups\". Theorem 2.1.
"},"LeinsterGroup.exists_nonabelian_leinster_group":{"url":"/FormalConjectures/Wikipedia/LeinsterGroup/#LeinsterGroup___exists_nonabelian_leinster_group","anchor":"LeinsterGroup___exists_nonabelian_leinster_group","docHtml":"\n Non-abelian Leinster groups exist.
\n\n For example, S₃ × C₅ (order 30) and A₅ × C₁₅₁₂₈ are Leinster groups.
\n Reference: Leinster, Tom (2001). \"Perfect numbers and groups\".
"},"LeinsterGroup.dihedral_is_leinster_iff_odd_perfect":{"url":"/FormalConjectures/Wikipedia/LeinsterGroup/#LeinsterGroup___dihedral_is_leinster_iff_odd_perfect","anchor":"LeinsterGroup___dihedral_is_leinster_iff_odd_perfect","docHtml":"\n The dihedral group DihedralGroup n (of order 2n) is a Leinster group if and only if n is\nan odd perfect number. This gives a one-to-one correspondence between dihedral Leinster groups\nand odd perfect numbers.
\n In particular, the existence of odd perfect numbers is equivalent to the existence of\ndihedral Leinster groups.
\n\n Reference: Leinster, Tom (2001). \"Perfect numbers and groups\".
"},"LittlewoodConjecture.littlewood_conjecture":{"url":"/FormalConjectures/Wikipedia/LittlewoodConjecture/#LittlewoodConjecture___littlewood_conjecture","anchor":"LittlewoodConjecture___littlewood_conjecture","docHtml":"\n For any two real numbers $\\alpha$ and $\\beta$,\n$$\n\\liminf_{n\\to\\infty} n||n\\alpha||||n\\beta|| = 0\n$$\nwhere $||x|| := \\min(|x - \\lfloor x \\rfloor|, |x - \\lceil x \\rceil|)$ is the distance\nto the nearest integer.
"},"LittlewoodConjecture.padic_littlewood_conjecture":{"url":"/FormalConjectures/Wikipedia/LittlewoodConjecture/#LittlewoodConjecture___padic_littlewood_conjecture","anchor":"LittlewoodConjecture___padic_littlewood_conjecture","docHtml":"\n For real number $\\alpha$ and prime $p$,\n$$\n\\liminf_{n \\to\\infty} n |n|
\n Euler's sum of powers conjecture states that for integers $n > 1$ and $k > 1$,\nif the sum of $n$ positive integers each raised to the $k$-th power equals another integer\nraised to the $k$-th power, then $n ≥ k$.
\n\n The conjecture is known to be false for $k = 4$ and $k = 5$,\nbut remains open for $k ≥ 6$.
"},"EulerSumOfPowers.eulers_sum_of_powers_conjecture.false_for_k4":{"url":"/FormalConjectures/Wikipedia/EulerSumOfPowers/#EulerSumOfPowers___eulers_sum_of_powers_conjecture___false_for_k4","anchor":"EulerSumOfPowers___eulers_sum_of_powers_conjecture___false_for_k4","docHtml":"\n Euler's sum of powers conjecture is false for $k=4$ (counterexample exists).
"},"EulerSumOfPowers.eulers_sum_of_powers_conjecture.false_for_k5":{"url":"/FormalConjectures/Wikipedia/EulerSumOfPowers/#EulerSumOfPowers___eulers_sum_of_powers_conjecture___false_for_k5","anchor":"EulerSumOfPowers___eulers_sum_of_powers_conjecture___false_for_k5","docHtml":"\n Euler's sum of powers conjecture is false for $k=5$ (counterexample exists).
"},"PrimesAndPerfectSquares.infinite_prime_sq_add_one":{"url":"/FormalConjectures/Wikipedia/PrimesAndPerfectSquares/#PrimesAndPerfectSquares___infinite_prime_sq_add_one","anchor":"PrimesAndPerfectSquares___infinite_prime_sq_add_one","docHtml":"\n Are there infinitely many primes $p$ such that $p - 1$ is a perfect square? In other words: Are there infinitely many primes of the form $n^2 + 1$?
"},"Selfridge.IsSelfridge":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___IsSelfridge","anchor":"Selfridge___IsSelfridge","docHtml":"\n A number p satisfies the
\np is odd,
\np ≡ ± 2 (mod 5),
\n2^(p-1) ≡ 1 (mod p)
\n(p+1).fib ≡ 0 (mod p)
\n This is the condition that is tested in the PSW conjecture.\nNote: this is non-standard terminology.
"},"Selfridge.IsSelfridge.is_odd":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___IsSelfridge___is_odd","anchor":"Selfridge___IsSelfridge___is_odd","docHtml":"\n A number p satisfies the
\np is odd,
\np ≡ ± 2 (mod 5),
\n2^(p-1) ≡ 1 (mod p)
\n(p+1).fib ≡ 0 (mod p)
\n This is the condition that is tested in the PSW conjecture.\nNote: this is non-standard terminology.
"},"Selfridge.IsSelfridge.mod_5":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___IsSelfridge___mod_5","anchor":"Selfridge___IsSelfridge___mod_5","docHtml":"\n A number p satisfies the
\np is odd,
\np ≡ ± 2 (mod 5),
\n2^(p-1) ≡ 1 (mod p)
\n(p+1).fib ≡ 0 (mod p)
\n This is the condition that is tested in the PSW conjecture.\nNote: this is non-standard terminology.
"},"Selfridge.IsSelfridge.pow_2":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___IsSelfridge___pow_2","anchor":"Selfridge___IsSelfridge___pow_2","docHtml":"\n A number p satisfies the
\np is odd,
\np ≡ ± 2 (mod 5),
\n2^(p-1) ≡ 1 (mod p)
\n(p+1).fib ≡ 0 (mod p)
\n This is the condition that is tested in the PSW conjecture.\nNote: this is non-standard terminology.
"},"Selfridge.IsSelfridge.fib":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___IsSelfridge___fib","anchor":"Selfridge___IsSelfridge___fib","docHtml":"\n A number p satisfies the
\np is odd,
\np ≡ ± 2 (mod 5),
\n2^(p-1) ≡ 1 (mod p)
\n(p+1).fib ≡ 0 (mod p)
\n This is the condition that is tested in the PSW conjecture.\nNote: this is non-standard terminology.
"},"Selfridge.IsPseudoSelfridge":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___IsPseudoSelfridge","anchor":"Selfridge___IsPseudoSelfridge","docHtml":"\n A number p satisfies the
\np is odd,
\np ≡ ± 1 (mod 5),
\n2^(p-1) ≡ 1 (mod p)
\n(p-1).fib ≡ 0 (mod p)
\n This is a variant of the condition that is tested in the PSW conjecture, and appears in the\nwiki page mentioned above.
\n\n Note: this is non-standard terminology.
"},"Selfridge.IsPseudoSelfridge.is_odd":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___IsPseudoSelfridge___is_odd","anchor":"Selfridge___IsPseudoSelfridge___is_odd","docHtml":"\n A number p satisfies the
\np is odd,
\np ≡ ± 1 (mod 5),
\n2^(p-1) ≡ 1 (mod p)
\n(p-1).fib ≡ 0 (mod p)
\n This is a variant of the condition that is tested in the PSW conjecture, and appears in the\nwiki page mentioned above.
\n\n Note: this is non-standard terminology.
"},"Selfridge.IsPseudoSelfridge.mod_5":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___IsPseudoSelfridge___mod_5","anchor":"Selfridge___IsPseudoSelfridge___mod_5","docHtml":"\n A number p satisfies the
\np is odd,
\np ≡ ± 1 (mod 5),
\n2^(p-1) ≡ 1 (mod p)
\n(p-1).fib ≡ 0 (mod p)
\n This is a variant of the condition that is tested in the PSW conjecture, and appears in the\nwiki page mentioned above.
\n\n Note: this is non-standard terminology.
"},"Selfridge.IsPseudoSelfridge.pow_2":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___IsPseudoSelfridge___pow_2","anchor":"Selfridge___IsPseudoSelfridge___pow_2","docHtml":"\n A number p satisfies the
\np is odd,
\np ≡ ± 1 (mod 5),
\n2^(p-1) ≡ 1 (mod p)
\n(p-1).fib ≡ 0 (mod p)
\n This is a variant of the condition that is tested in the PSW conjecture, and appears in the\nwiki page mentioned above.
\n\n Note: this is non-standard terminology.
"},"Selfridge.IsPseudoSelfridge.fib":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___IsPseudoSelfridge___fib","anchor":"Selfridge___IsPseudoSelfridge___fib","docHtml":"\n A number p satisfies the
\np is odd,
\np ≡ ± 1 (mod 5),
\n2^(p-1) ≡ 1 (mod p)
\n(p-1).fib ≡ 0 (mod p)
\n This is a variant of the condition that is tested in the PSW conjecture, and appears in the\nwiki page mentioned above.
\n\n Note: this is non-standard terminology.
"},"Selfridge.selfridge_conjecture":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___selfridge_conjecture","anchor":"Selfridge___selfridge_conjecture","docHtml":"\nPSW conjecture (Selfridge's test)\nLet $p$ be an odd number, with $p \\equiv \\pm 2 \\pmod{5}$, $2^{p-1} \\equiv 1 \\pmod{p}$\nand $F_{p+1} \\equiv 0 \\pmod{p}$, then $p$ is a prime number.
"},"Selfridge.selfridge_conjecture.variants.exist_pseudo_counterexample":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___selfridge_conjecture___variants___exist_pseudo_counterexample","anchor":"Selfridge___selfridge_conjecture___variants___exist_pseudo_counterexample","docHtml":"\n Selfridge's test variant:\nLet $p$ be an odd number, with $p \\equiv \\pm 1 \\pmod{5}$, $2^{p-1} \\equiv 1 \\pmod{p}$\nand $F_{p-1} \\equiv 0 \\pmod{p}$, then $p$ is a prime number.
\n\n This test does not work.
"},"Selfridge.selfridge_conjecture.variants.pseudo_counterexample":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___selfridge_conjecture___variants___pseudo_counterexample","anchor":"Selfridge___selfridge_conjecture___variants___pseudo_counterexample","docHtml":"\n Selfridge's test variant:\nLet $p$ be an odd number, with $p \\equiv \\pm 1 \\pmod{5}$, $2^{p-1} \\equiv 1 \\pmod{p}$\nand $F_{p-1} \\equiv 0 \\pmod{p}$, then $p$ is a prime number.
\n\n The number $6601$ is a conterexample to this test satisfying $6601 ≡ 1 \\mod 5$
"},"Selfridge.selfridge_conjecture.variants.pseudo_counterexample'":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___selfridge_conjecture___variants___pseudo_counterexample___","anchor":"Selfridge___selfridge_conjecture___variants___pseudo_counterexample___","docHtml":"\n Selfridge's test variant:\nLet $p$ be an odd number, with $p \\equiv \\pm 1 \\pmod{5}$, $2^{p-1} \\equiv 1 \\pmod{p}$\nand $F_{p-1} \\equiv 0 \\pmod{p}$, then $p$ is a prime number.
\n\n The number $30889$ is a conterexample to this test satisfying $30889 ≡ - 1 \\mod 5$
"},"Selfridge.fermatFactors":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___fermatFactors","anchor":"Selfridge___fermatFactors","docHtml":"\nOEIS A46052\nThe number of distinct prime factors of nth Fermat number.\nKnown terms: 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5
"},"Selfridge.selfridge_seq_conjecture":{"url":"/FormalConjectures/Wikipedia/Selfridge/#Selfridge___selfridge_seq_conjecture","anchor":"Selfridge___selfridge_seq_conjecture","docHtml":"\n Selfridge conjectured that the number of prime factors of the n-th Fermat number does not grow\nmonotonically in $n$.
\n Selfridge conjectured that the number of prime factors of the n-th Fermat number does not grow\nmonotonically in $n$.
\n A sufficient condition for this conjecture to hold is that there exists a Fermat prime larger than\n65537.
"},"KummerVandiver.kummer_vandiver":{"url":"/FormalConjectures/Wikipedia/KummerVandiver/#KummerVandiver___kummer_vandiver","anchor":"KummerVandiver___kummer_vandiver","docHtml":"\n
\n The standard class $\\mathcal{S}$ of normalised univalent functions on $\\mathbb{D}$.
"},"BrennanConjecture.IsUnivalentNormalized.analyticOn":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___IsUnivalentNormalized___analyticOn","anchor":"BrennanConjecture___IsUnivalentNormalized___analyticOn","docHtml":"\n The standard class $\\mathcal{S}$ of normalised univalent functions on $\\mathbb{D}$.
"},"BrennanConjecture.IsUnivalentNormalized.injOn":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___IsUnivalentNormalized___injOn","anchor":"BrennanConjecture___IsUnivalentNormalized___injOn","docHtml":"\n The standard class $\\mathcal{S}$ of normalised univalent functions on $\\mathbb{D}$.
"},"BrennanConjecture.IsUnivalentNormalized.map_zero":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___IsUnivalentNormalized___map_zero","anchor":"BrennanConjecture___IsUnivalentNormalized___map_zero","docHtml":"\n The standard class $\\mathcal{S}$ of normalised univalent functions on $\\mathbb{D}$.
"},"BrennanConjecture.IsUnivalentNormalized.deriv_zero":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___IsUnivalentNormalized___deriv_zero","anchor":"BrennanConjecture___IsUnivalentNormalized___deriv_zero","docHtml":"\n The standard class $\\mathcal{S}$ of normalised univalent functions on $\\mathbb{D}$.
"},"BrennanConjecture.integralMeansSpectrum":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___integralMeansSpectrum","anchor":"BrennanConjecture___integralMeansSpectrum","docHtml":"\n $\\beta_f(\\tau) := \\limsup_{r \\to 1^-}\n\\frac{\\log \\int_{-\\pi}^{\\pi} |f'(re^{i\\theta})|^\\tau , d\\theta}{|\\log(1-r)|}$
"},"BrennanConjecture.universalSpectrum":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___universalSpectrum","anchor":"BrennanConjecture___universalSpectrum"},"BrennanConjecture.universalSpectrumBounded":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___universalSpectrumBounded","anchor":"BrennanConjecture___universalSpectrumBounded"},"BrennanConjecture.universalSpectrumBounded_le":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___universalSpectrumBounded_le","anchor":"BrennanConjecture___universalSpectrumBounded_le"},"BrennanConjecture.integralMeansSpectrum_id":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___integralMeansSpectrum_id","anchor":"BrennanConjecture___integralMeansSpectrum_id"},"BrennanConjecture.brennan_universalSpectrum":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___brennan_universalSpectrum","anchor":"BrennanConjecture___brennan_universalSpectrum","docHtml":"\n Brennan's conjecture, part 1: $B(-2) = 1$.
"},"BrennanConjecture.brennan_universalSpectrumBounded":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___brennan_universalSpectrumBounded","anchor":"BrennanConjecture___brennan_universalSpectrumBounded","docHtml":"\n Brennan's conjecture, part 2: $B_b(-2) = 1$.
"},"BrennanConjecture.brennan_spectra_eq":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___brennan_spectra_eq","anchor":"BrennanConjecture___brennan_spectra_eq","docHtml":"\n Brennan's conjecture, part 3: $B(-2) = B_b(-2)$.
"},"BrennanConjecture.brennan":{"url":"/FormalConjectures/Wikipedia/Brennanconjecture/#BrennanConjecture___brennan","anchor":"BrennanConjecture___brennan","docHtml":"\n Brennan's conjecture: $B(-2) = B_b(-2) = 1$.
"},"ModularityConjecture.modularFormAn":{"url":"/FormalConjectures/Wikipedia/ModularityConjecture/#ModularityConjecture___modularFormAn","anchor":"ModularityConjecture___modularFormAn","docHtml":"\n The n-th Fourier coefficient of a modular forms (around the cusp at infinity).
\n We need to reduce a rational modulo p, in practice we wont be dividing by zero since the\nconductor of the elliptic curve saves us.
\n The set of points on an elliptic curve over ZMod n.
\n The set of point mod n is finite.
\n Note that normally this is written as p + 1 - #E(𝔽ₚ), but since we don't have a point at\ninfinty on this affine curve we only have p
\n Since we don't have Hecke operators yet, we define this via the q-expansion coefficients. See\nProposition 5.8.5 of [diamondshurman2005].
"},"ModularityConjecture.modularityConjecture":{"url":"/FormalConjectures/Wikipedia/ModularityConjecture/#ModularityConjecture___modularityConjecture","anchor":"ModularityConjecture___modularityConjecture","docHtml":"\n See theorem 8.8.1 of [diamondshurman2005].
"},"ModularityConjecture.modularity_conjecture":{"url":"/FormalConjectures/Wikipedia/ModularityConjecture/#ModularityConjecture___modularity_conjecture","anchor":"ModularityConjecture___modularity_conjecture","docHtml":"\n The Modularity Theorem (formerly Shimura-Taniyama-Weil conjecture): every elliptic curve\nover $\\mathbb{Q}$ is modular.
"},"WoodallPrimes.infinitely_many_woodall_primes":{"url":"/FormalConjectures/Wikipedia/WoodalPrimes/#WoodallPrimes___infinitely_many_woodall_primes","anchor":"WoodallPrimes___infinitely_many_woodall_primes","docHtml":"\n There are infinitely many prime numbers of the form k * 2 ^ k - 1 for k > 1.
\nIsBuchi M asserts that whenever M consecutive values (x + n)² + a (for\nn = 0, …, M - 1) are all perfect squares, then a must be 0.
\nBüchi's problem\nThere exists a positive integer $M$ such that, for all integers $x$ and $a$,\nif $(x+n)^2 + a$ is a square for $M$ consecutive values of $n$, then $a = 0$.
"},"Buchi.buchi_problem_M5":{"url":"/FormalConjectures/Wikipedia/Buchi/#Buchi___buchi_problem_M5","anchor":"Buchi___buchi_problem_M5","docHtml":"\nBüchi's problem (first open case, $M = 5$):\nFor all integers $x$ and $a$, if $(x+n)^2 + a$ is a perfect square for $n = 0, 1, 2, 3, 4$,\nthen $a = 0$.
\n\n Non-trivial sequences of length 3 and 4 are known to exist, so $M = 5$ is the first open case.
"},"Buchi.buchi_false_M0":{"url":"/FormalConjectures/Wikipedia/Buchi/#Buchi___buchi_false_M0","anchor":"Buchi___buchi_false_M0","docHtml":"\n The case $M = 0$ fails: the hypothesis is vacuous, so $a = 0$ cannot be concluded.
"},"Buchi.buchi_false_M1":{"url":"/FormalConjectures/Wikipedia/Buchi/#Buchi___buchi_false_M1","anchor":"Buchi___buchi_false_M1","docHtml":"\n The case $M = 1$ fails: $0^2 + 4 = 4 = 2^2$ is a square but $a = 4 \\neq 0$.
"},"Buchi.buchi_false_M2":{"url":"/FormalConjectures/Wikipedia/Buchi/#Buchi___buchi_false_M2","anchor":"Buchi___buchi_false_M2","docHtml":"\n The case $M = 2$ fails: $(7+n)^2 - 48$ is a perfect square for $n = 0, 1$\n($1^2$ and $4^2$), but $a = -48 \\neq 0$.
"},"Buchi.buchi_false_M3":{"url":"/FormalConjectures/Wikipedia/Buchi/#Buchi___buchi_false_M3","anchor":"Buchi___buchi_false_M3","docHtml":"\n The case $M = 3$ fails: $(24+n)^2 - 576$ is a perfect square for $n = 0, 1, 2$\n($0^2$, $7^2$, and $10^2$), but $a = -576 \\neq 0$.
"},"Buchi.buchi_false_M4":{"url":"/FormalConjectures/Wikipedia/Buchi/#Buchi___buchi_false_M4","anchor":"Buchi___buchi_false_M4","docHtml":"\n The case $M = 4$ fails: $(246+n)^2 - 60480$ is a perfect square for $n = 0, 1, 2, 3$\n($6^2$, $23^2$, $32^2$, and $39^2$), but $a = -60480 \\neq 0$.\nThe sequence $(6, 23, 32, 39)$ is a nontrivial Büchi sequence of length 4 (Hensley 1983).
"},"PellNumbers.pellNumber":{"url":"/FormalConjectures/Wikipedia/Pell/#PellNumbers___pellNumber","anchor":"PellNumbers___pellNumber","docHtml":"\n The
\n Similar to Fibonacci numbers, there exist numerous identites around Pell numbers, i.e.\nP_{2n+1} = P_n ^ 2 + P_{n+1} ^ 2
"},"PellNumbers.coe_pellNumber_eq":{"url":"/FormalConjectures/Wikipedia/Pell/#PellNumbers___coe_pellNumber_eq","anchor":"PellNumbers___coe_pellNumber_eq","docHtml":"\n An explicit formula for Pell numbers, similar to Binet's formula
"},"PellNumbers.infinite_pellNumber_primes":{"url":"/FormalConjectures/Wikipedia/Pell/#PellNumbers___infinite_pellNumber_primes","anchor":"PellNumbers___infinite_pellNumber_primes","docHtml":"\n There are infinitely many prime Pell numbers
"},"Fuglede.FugledeConjectureFor":{"url":"/FormalConjectures/Wikipedia/Fuglede/#Fuglede___FugledeConjectureFor","anchor":"Fuglede___FugledeConjectureFor","docHtml":"\nFuglede's conjecture in dimension n: A bounded subset of ℝ^n with positive Lebesgue measure is spectral iff it tiles ℝ^n by translation.
\nFuglede's conjecture in one dimension: A bounded subset of ℝ with positive Lebesgue measure is spectral iff it tiles ℝ by translation.
"},"Fuglede.FugledeConjecture.variants.dim_2":{"url":"/FormalConjectures/Wikipedia/Fuglede/#Fuglede___FugledeConjecture___variants___dim_2","anchor":"Fuglede___FugledeConjecture___variants___dim_2","docHtml":"\nFuglede's conjecture in two dimensions: A bounded subset of ℝ^2 with positive Lebesgue measure is spectral iff it tiles ℝ^2 by translation.
"},"Fuglede.FugledeConjecture.variants.dim_3_or_higher":{"url":"/FormalConjectures/Wikipedia/Fuglede/#Fuglede___FugledeConjecture___variants___dim_3_or_higher","anchor":"Fuglede___FugledeConjecture___variants___dim_3_or_higher","docHtml":"\nFuglede's conjecture in three or higher dimensions has been disproven.\n(Note that counterexamples in lower dimensions would also disprove the conjecture in higher dimensions.)
"},"Toronto.TorontoSpace":{"url":"/FormalConjectures/Wikipedia/Toronto/#Toronto___TorontoSpace","anchor":"Toronto___TorontoSpace","docHtml":"\n A
\n A
\n Every finite space is Toronto, since\nthe only subspace with same cardinality is the space itself.
"},"Toronto.DiscreteTopology.of_t2_of_torontoSpace":{"url":"/FormalConjectures/Wikipedia/Toronto/#Toronto___DiscreteTopology___of_t2_of_torontoSpace","anchor":"Toronto___DiscreteTopology___of_t2_of_torontoSpace","docHtml":"\n Any T2, Toronto space is discrete.
"},"VaughtConjecture.numberOfCountableModels":{"url":"/FormalConjectures/Wikipedia/VaughtConjecture/#VaughtConjecture___numberOfCountableModels","anchor":"VaughtConjecture___numberOfCountableModels","docHtml":"\n The number of countable models of some L-Theory T up to isomorphism
"},"VaughtConjecture.vaught_conjecture":{"url":"/FormalConjectures/Wikipedia/VaughtConjecture/#VaughtConjecture___vaught_conjecture","anchor":"VaughtConjecture___vaught_conjecture","docHtml":"\n The Vaught conjecture states that for a countable language L and a complete L-Theory T\nthe number of countable models of T (up to isomorphism) is finite, $\\aleph_0$ or $2^{\\aleph_0}$.
"},"NormalNumber.pi_normal_base_ten":{"url":"/FormalConjectures/Wikipedia/NormalityOfPi/#NormalNumber___pi_normal_base_ten","anchor":"NormalNumber___pi_normal_base_ten","docHtml":"\n $\\pi$ is normal in base 10.
"},"BalancedPrimes.balanced_primes":{"url":"/FormalConjectures/Wikipedia/BalancedPrimes/#BalancedPrimes___balanced_primes","anchor":"BalancedPrimes___balanced_primes","docHtml":"\n Let $p_k$ be the $k$-th prime number.\nAre there infinitely many $n$ such that $(p_n + p_{n+2}) / 2$ is prime?
"},"BalancedPrimes.balanced_primes_order":{"url":"/FormalConjectures/Wikipedia/BalancedPrimes/#BalancedPrimes___balanced_primes_order","anchor":"BalancedPrimes___balanced_primes_order","docHtml":"\n Let $p_k$ be the $k$-th prime number.\nAre there infinitely many $n$ such that\n$p_n = \\dfrac{\\sum_{i = 1} ^ k p_{n - i} + p_{n + i}}{2*k}$?
"},"ArtinPrimitiveRootsConjecture.S":{"url":"/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/#ArtinPrimitiveRootsConjecture___S","anchor":"ArtinPrimitiveRootsConjecture___S","docHtml":"\n Let $S(a)$ be the set of primes such that $a$ is a primitive root modulo $p$.
"},"ArtinPrimitiveRootsConjecture.ArtinConstant":{"url":"/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/#ArtinPrimitiveRootsConjecture___ArtinConstant","anchor":"ArtinPrimitiveRootsConjecture___ArtinConstant","docHtml":"\nArtin's Constant is defined to be the product\n$$\\prod_{p\\ \\text{prime}} \\left(1 - \\frac{1}{p(p - 1)}\\right)$$.
"},"ArtinPrimitiveRootsConjecture.powCorrectionFactor":{"url":"/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/#ArtinPrimitiveRootsConjecture___powCorrectionFactor","anchor":"ArtinPrimitiveRootsConjecture___powCorrectionFactor","docHtml":"\n Artin's conjecture on $S(a)$ when $a = b^m$ is a power, where $m$ is odd and maximal,\nrequires a correction factor to multiply ArtinConstant and is given by\n$$\\prod_{p \\mid m} \\frac{p(p - 2)}{p^2 - p - 1}.$$
\n Artin's conjecture on $S(a)$ when $a = b^m$ is a power, and the squarefree part\nof $b_0\\equiv 1\\pmod{4}$, requires a further correct factor to\nArtinConstant * powCorrectionFactor m, which modifies primes which divide\n$\\gcd(b_0, m)$ and primes which do not divide $m$ separately as\n$$ 1 - \\prod_{p \\mid \\gcd(b_0, m)} \\frac{1}{2 - p}\n\\prod_{p \\mid b_0, p\\nmid m} \\frac{1}{1 + p - p^2}.$$
\nArtin's Conjecture on Primitive Roots, first half.\nLet $a$ be an integer that is not a square number and not $−1$. Then the set $S(a)$\nof primes $p$ such that $a$ is a primitive root modulo $p$ has a positive asymptotic\ndensity inside the set of primes. In particular, $S(a)$ is infinite.
"},"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.parts.i":{"url":"/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/#ArtinPrimitiveRootsConjecture___conditional_artin_primitive_roots___parts___i","anchor":"ArtinPrimitiveRootsConjecture___conditional_artin_primitive_roots___parts___i","docHtml":"\nArtin's Conjecture on Primitive Roots, first half, conditional on GRH.
"},"ArtinPrimitiveRootsConjecture.artin_primitive_roots.parts.ii":{"url":"/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/#ArtinPrimitiveRootsConjecture___artin_primitive_roots___parts___ii","anchor":"ArtinPrimitiveRootsConjecture___artin_primitive_roots___parts___ii","docHtml":"\nArtin's Conjecture on Primitive Roots, second half.\nWrite $a = a_0 b^2$ where $a_0$ is squarefree. Under the conditions that $a$ is not a perfect\npower and $a_0\\not\\equiv 1\\pmod{4}$ (sequence A85397 in the OEIS), the density of the set\n$S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$ is independent of $a$ and\nequals Artin's constant.
"},"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.parts.ii":{"url":"/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/#ArtinPrimitiveRootsConjecture___conditional_artin_primitive_roots___parts___ii","anchor":"ArtinPrimitiveRootsConjecture___conditional_artin_primitive_roots___parts___ii","docHtml":"\nArtin's Conjecture on Primitive Roots, second half, conditional on GRH.
"},"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_square_or_minus_one":{"url":"/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/#ArtinPrimitiveRootsConjecture___artin_primitive_roots___variants___part_ii_square_or_minus_one","anchor":"ArtinPrimitiveRootsConjecture___artin_primitive_roots___variants___part_ii_square_or_minus_one","docHtml":"\nArtin's Conjecture on Primitive Roots, second half, different residue version\nIf $a$ is a square number or $a = −1$, then the density of the set $S(a)$ of primes\n$p$ such that $a$ is a primitive root modulo $p$ is $0$.
"},"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_power_squarefreePart_not_modeq_one":{"url":"/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/#ArtinPrimitiveRootsConjecture___artin_primitive_roots___variants___part_ii_power_squarefreePart_not_modeq_one","anchor":"ArtinPrimitiveRootsConjecture___artin_primitive_roots___variants___part_ii_power_squarefreePart_not_modeq_one","docHtml":"\nArtin's Conjecture on Primitive Roots, second half, power version\nIf $a = b^m$ is a perfect odd power of a number $b$ whose squarefree part\n$b_0\\not\\equiv 1 \\pmod{4}$, then the density of the set $S(a)$ of primes $p$ such that\n$a$ is a primitive root modulo $p$ is given by\n$$C\\prod_{p \\mid m} \\frac{p(p - 2)}{p^2 - p - 1}$$,\nwhere $C$ is Artin's constant.
"},"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.variants.part_ii_power_squarefreePart_not_modeq_one":{"url":"/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/#ArtinPrimitiveRootsConjecture___conditional_artin_primitive_roots___variants___part_ii_power_squarefreePart_not_modeq_one","anchor":"ArtinPrimitiveRootsConjecture___conditional_artin_primitive_roots___variants___part_ii_power_squarefreePart_not_modeq_one","docHtml":"\nArtin's Conjecture on Primitive Roots, second half, power version, conditional on GRH
"},"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one":{"url":"/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/#ArtinPrimitiveRootsConjecture___artin_primitive_roots___variants___part_ii_power_squarefreePart_modeq_one","anchor":"ArtinPrimitiveRootsConjecture___artin_primitive_roots___variants___part_ii_power_squarefreePart_modeq_one","docHtml":"\nArtin's Conjecture on Primitive Roots, second half, power version\nIf $a = b^m$ is a perfect power of a number $b$ whose squarefree part $b_0\\equiv 1 \\pmod{4}$,\nthen the density of the set $S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$\nis given by\n$$C \\left(\\prod_{p \\mid m} \\frac{p(p-2)}{(p ^ 2 - p - 1)}\\right)\n\\left(1 - \\prod_{p \\mid \\gcd(b_0, m)} \\frac{1}{2 - p}\n\\prod_{p \\mid b_0, p\\nmid m} \\frac{1}{(1 + p - p ^ 2)}\\right),$$\nwhere $C$ is Artin's constant.
"},"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one":{"url":"/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/#ArtinPrimitiveRootsConjecture___conditional_artin_primitive_roots___variants___part_ii_power_squarefreePart_modeq_one","anchor":"ArtinPrimitiveRootsConjecture___conditional_artin_primitive_roots___variants___part_ii_power_squarefreePart_modeq_one","docHtml":"\nArtin's Conjecture on Primitive Roots, second half, power version, conditional on GRH.
"},"PebblingNumberConjecture.PebbleDistribution":{"url":"/FormalConjectures/Wikipedia/PebblingNumberConjecture/#PebblingNumberConjecture___PebbleDistribution","anchor":"PebblingNumberConjecture___PebbleDistribution","docHtml":"\n A Pebble distribution is an assigment of zero or more pebbles to each of the vertices.
"},"PebblingNumberConjecture.NumberOfPebbles":{"url":"/FormalConjectures/Wikipedia/PebblingNumberConjecture/#PebblingNumberConjecture___NumberOfPebbles","anchor":"PebblingNumberConjecture___NumberOfPebbles","docHtml":"\n The number of pebbles of a distribution is the total number summed over all vertices.
"},"PebblingNumberConjecture.IsPebblingMove":{"url":"/FormalConjectures/Wikipedia/PebblingNumberConjecture/#PebblingNumberConjecture___IsPebblingMove","anchor":"PebblingNumberConjecture___IsPebblingMove","docHtml":"\n A pebbling move on a graph consists of choosing a vertex with at least two pebbles, removing\ntwo pebbles from it, and adding one to an adjacent vertex (the second removed pebble is discarded\nfrom play).
"},"PebblingNumberConjecture.IsPebblingMove.refl":{"url":"/FormalConjectures/Wikipedia/PebblingNumberConjecture/#PebblingNumberConjecture___IsPebblingMove___refl","anchor":"PebblingNumberConjecture___IsPebblingMove___refl"},"PebblingNumberConjecture.PebblePath":{"url":"/FormalConjectures/Wikipedia/PebblingNumberConjecture/#PebblingNumberConjecture___PebblePath","anchor":"PebblingNumberConjecture___PebblePath","docHtml":"\n A pebble path is a series of pebbling moves.
"},"PebblingNumberConjecture.PebblePath.refl":{"url":"/FormalConjectures/Wikipedia/PebblingNumberConjecture/#PebblingNumberConjecture___PebblePath___refl","anchor":"PebblingNumberConjecture___PebblePath___refl","docHtml":"\n A pebble path is a series of pebbling moves.
"},"PebblingNumberConjecture.PebblePath.step":{"url":"/FormalConjectures/Wikipedia/PebblingNumberConjecture/#PebblingNumberConjecture___PebblePath___step","anchor":"PebblingNumberConjecture___PebblePath___step","docHtml":"\n A pebble path is a series of pebbling moves.
"},"PebblingNumberConjecture.ExistsPebblePath":{"url":"/FormalConjectures/Wikipedia/PebblingNumberConjecture/#PebblingNumberConjecture___ExistsPebblePath","anchor":"PebblingNumberConjecture___ExistsPebblePath","docHtml":"\n Indicates whether there exists a sequence of pebbling moves transforming one pebble distribution\nto another.
"},"PebblingNumberConjecture.IsReachable":{"url":"/FormalConjectures/Wikipedia/PebblingNumberConjecture/#PebblingNumberConjecture___IsReachable","anchor":"PebblingNumberConjecture___IsReachable","docHtml":"\n A pebble distribution B is reachable from another pebble distribution A, if there exists a\nsequence of pebbling moves transforming the first into the second.
\n The pebbling number of a graph G, is the lowest natural number n that satisfies the\nfollowing condition: Given any target or 'root' vertex in the graph and any initial\npebbles distribution with n pebbles on the graph, another pebble distribution is reachable\nin which the designated root vertex has one or more pebbles.
\n The pebbling number of the complete graph on n vertices is n.
\n The pebbling number conjecture:\nthe pebbling number of a Cartesian product of graphs is at most equal to the product of the\npebbling numbers of the factors.
"},"Singmaster.solutions":{"url":"/FormalConjectures/Wikipedia/Singmaster/#Singmaster___solutions","anchor":"Singmaster___solutions","docHtml":"\n The set of pairs (n, k) representing the solutions to the equation\nNat.choose n k = t for a given t, under the constraint 1 ≤ k < n.
\n Singmaster's conjecture: the number of times any number $t > 1$ appears in\nPascal's triangle is bounded.
"},"Superperfect.PerfectFor":{"url":"/FormalConjectures/Wikipedia/Superperfectnumbers/#Superperfect___PerfectFor","anchor":"Superperfect___PerfectFor","docHtml":"\n A positive integer $n$ is $(m,k)$-perfect if $\\sigma^m(n) = kn$ where $\\sigma^m$ is the $m$-th iterate of $σ$.
"},"Superperfect.twoFivePerfect":{"url":"/FormalConjectures/Wikipedia/Superperfectnumbers/#Superperfect___twoFivePerfect","anchor":"Superperfect___twoFivePerfect","docHtml":"\n There does not exist a $(2,5)$-perfect number
"},"Schanuel.schanuel_conjecture":{"url":"/FormalConjectures/Wikipedia/Schanuel/#Schanuel___schanuel_conjecture","anchor":"Schanuel___schanuel_conjecture","docHtml":"\n Given any set of $n$ complex numbers ${z_1, ..., z_n}$ that are linearly independent over\n$\\mathbb{Q}$, the field extension $\\mathbb{Q}(z_1, ..., z_n, e^{z_1}, ..., e^{z_n})$\nhas transcendence degree at least $n$ over $\\mathbb{Q}$.
"},"RationalDistanceProblem.UnitSquareCorners":{"url":"/FormalConjectures/Wikipedia/RationalDistanceProblem/#RationalDistanceProblem___UnitSquareCorners","anchor":"RationalDistanceProblem___UnitSquareCorners","docHtml":"\n
\nmathoverflow/418260\nasked by user Yuan Yang
\n\n D19 in Unsolved Problems in Number Theory\nby
\n Does there exist a point in the plane at rational distance from all four vertices of the unit square?
"},"Brocard.brocard_conjecture":{"url":"/FormalConjectures/Wikipedia/BrocardConjecture/#Brocard___brocard_conjecture","anchor":"Brocard___brocard_conjecture","docHtml":"\nBrocard's Conjecture\nFor every n ≥ 2, between the squares of the n-th and (n+1)-th primes,\nthere are at least four prime numbers.
\n Ferreira proved that Brocard's conjecture is true for sufficiently large n.
"},"GoldbachConjecture.goldbach":{"url":"/FormalConjectures/Wikipedia/GoldbachConjecture/#GoldbachConjecture___goldbach","anchor":"GoldbachConjecture___goldbach","docHtml":"\n Can every even integer greater than 2 be written as the sum of two primes?
"},"TernaryGoldbachConjecture.ternaryGoldbach":{"url":"/FormalConjectures/Wikipedia/GoldbachConjecture/#TernaryGoldbachConjecture___ternaryGoldbach","anchor":"TernaryGoldbachConjecture___ternaryGoldbach","docHtml":"\n Can every odd integer greater than 5 be written as the sum of three primes?\n(A prime may be used more than once.)
\n\n NB. While Harald Helfgott's solution is not published in a peer-reviewed journal yet,\nhis results seem generally accepted.
"},"DedekindNumber.piFinBoolDecidableLE":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___piFinBoolDecidableLE","anchor":"DedekindNumber___piFinBoolDecidableLE","docHtml":"\n A Dedekind number M(n) counts the number of monotone Boolean functions on n variables,\nor equivalently, the number of antichains (Sperner families) in the Boolean lattice 2^[n].
\n For example,\n$$M ( 0 ) = 2 , M ( 1 ) = 3 , M ( 2 ) = 6 , and M ( 3 ) = 20 .$$\nThe first few values grew slowly:\n$$M ( 4 ) = 168 , M ( 5 ) = 7581$$,\nbut then rapidly:\n$$M ( 6 ) = 7828354 , M ( 7 ) = 2414682040998 , M ( 8 ) = 56130437228687557907788$$, and\n$$M ( 9 ) = 286386577668298411128469151667598498812366$$\n(computed in 2023).
\n\n We formalize two definitions:
\n\nM n: the number of monotone Boolean functions (Fin n → Bool) → Bool
\nM' n: the number of antichains (Sperner families) of Finset (Fin n)
\n We prove their values for small n and show that the two definitions agree for all n.
\n The problem is to determine the exact values of $M(n)$ for $n ≥ 10$.\nIn particular, the value of $M(10)$ is currently unknown.
\n\n
\n $M(n)$ is the number of monotone Boolean functions on $n$ variables.
"},"DedekindNumber.IsSperner":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___IsSperner","anchor":"DedekindNumber___IsSperner","docHtml":"\n A Sperner family (antichain) of subsets of Fin n: a family of sets such that\nno member is a subset of another.
\n $M'(n)$ is the number of antichains (Sperner families) of subsets of Fin n.
\n Values for small n
"},"DedekindNumber.M_one":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___M_one","anchor":"DedekindNumber___M_one"},"DedekindNumber.M_two":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___M_two","anchor":"DedekindNumber___M_two"},"DedekindNumber.M_three":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___M_three","anchor":"DedekindNumber___M_three"},"DedekindNumber.M'_zero":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___M____zero","anchor":"DedekindNumber___M____zero"},"DedekindNumber.M'_one":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___M____one","anchor":"DedekindNumber___M____one"},"DedekindNumber.M'_two":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___M____two","anchor":"DedekindNumber___M____two"},"DedekindNumber.M'_three":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___M____three","anchor":"DedekindNumber___M____three"},"DedekindNumber.χ":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber______","anchor":"DedekindNumber______","docHtml":"\n The indicator function of a finset: χ s i = true ↔ i ∈ s.
\n The support of a Boolean-valued function: supp v = {i | v i = true}.
\n Forward map: monotone function → Sperner family (the minimal true sets).
"},"DedekindNumber.fromSperner":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___fromSperner","anchor":"DedekindNumber___fromSperner","docHtml":"\n Backward map: Sperner family → monotone Boolean function.
"},"DedekindNumber.χ_supp":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber_______supp","anchor":"DedekindNumber_______supp"},"DedekindNumber.supp_χ":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___supp____","anchor":"DedekindNumber___supp____"},"DedekindNumber.χ_le_iff":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber_______le_iff","anchor":"DedekindNumber_______le_iff"},"DedekindNumber.mem_supp_iff":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___mem_supp_iff","anchor":"DedekindNumber___mem_supp_iff"},"DedekindNumber.toSperner_isSperner":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___toSperner_isSperner","anchor":"DedekindNumber___toSperner_isSperner"},"DedekindNumber.fromSperner_monotone":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___fromSperner_monotone","anchor":"DedekindNumber___fromSperner_monotone"},"DedekindNumber.exists_minimal_true_subset":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___exists_minimal_true_subset","anchor":"DedekindNumber___exists_minimal_true_subset","docHtml":"\n Every true set of a monotone Boolean function contains a minimal true set.
"},"DedekindNumber.fromSperner_toSperner":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___fromSperner_toSperner","anchor":"DedekindNumber___fromSperner_toSperner","docHtml":"\n Converting a monotone function to a Sperner family and back yields the same function.
"},"DedekindNumber.toSperner_fromSperner":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___toSperner_fromSperner","anchor":"DedekindNumber___toSperner_fromSperner","docHtml":"\n Converting a Sperner family to a monotone function and back yields the same family.
"},"DedekindNumber.equivMonotoneSperner":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___equivMonotoneSperner","anchor":"DedekindNumber___equivMonotoneSperner","docHtml":"\n The set of monotone Boolean functions on n variables is in bijection\nwith the set of Sperner families of subsets of Fin n.
\n A closed formula for the Dedekind numbers as found by Kisielewicz (1998):\n$$\nM(n) = \\sum_{k=0}^{2^{2^n}}\\prod_{j = 1}^{2 ^ n - 1}\\prod_{i = 0}^{j - 1} \\left(\n1 - b_i^kb_j^k \\prod_{m = 0}^{\\log_2 i} (1 - b_m^i + b_m^ib_m^j)\\right),\n$$,\nwhere $b_i^k$ is the $i$-th bit of $k$.
"},"DedekindNumber.kisielewiczFormula_zero":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___kisielewiczFormula_zero","anchor":"DedekindNumber___kisielewiczFormula_zero"},"DedekindNumber.kisielewiczFormula_one":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___kisielewiczFormula_one","anchor":"DedekindNumber___kisielewiczFormula_one"},"DedekindNumber.kisielewiczFormula_two":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___kisielewiczFormula_two","anchor":"DedekindNumber___kisielewiczFormula_two"},"DedekindNumber.kisielewiczFormula_three":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___kisielewiczFormula_three","anchor":"DedekindNumber___kisielewiczFormula_three"},"DedekindNumber.M_eq_kisielewiczFormula":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___M_eq_kisielewiczFormula","anchor":"DedekindNumber___M_eq_kisielewiczFormula","docHtml":"\n Kisielewicz (1988) proved the following arithmetic formula for the Dedekind numbers:\n$$\nM(n) = \\sum_{k=0}^{2^{2^n}}\\prod_{j = 1}^{2 ^ n - 1}\\prod_{i = 0}^{j - 1} \\left(\n1 - b_i^kb_j^k \\prod_{m = 0}^{\\log_2 i} (1 - b_m^i + b_m^ib_m^j)\\right),\n$$\nwhere $b_i^k$ is the $i$-th bit of $k$. However, this formula is not computationally\nefficient for large $n$.
"},"DedekindNumber.M_eq":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___M_eq","anchor":"DedekindNumber___M_eq","docHtml":"\n No closed-form expression that allows efficient computation of Dedekind numbers is\ncurrently known.
"},"DedekindNumber.Dedekind_10":{"url":"/FormalConjectures/Wikipedia/DedekindNumber/#DedekindNumber___Dedekind_10","anchor":"DedekindNumber___Dedekind_10","docHtml":"\n In particular, the Dedekind number for n = 10 is currently unknown.
\n Natural number $n$ for which there exists a $m ≠ n$ with $φ(m) = φ(n)$
"},"CarmichaelTotient.carchimichealTotientFor_zero":{"url":"/FormalConjectures/Wikipedia/CarmichaelTotient/#CarmichaelTotient___carchimichealTotientFor_zero","anchor":"CarmichaelTotient___carchimichealTotientFor_zero","docHtml":"\n $n = 0 ↔ φ(n) = 0$
"},"CarmichaelTotient.carmichealTotientFor_odd":{"url":"/FormalConjectures/Wikipedia/CarmichaelTotient/#CarmichaelTotient___carmichealTotientFor_odd","anchor":"CarmichaelTotient___carmichealTotientFor_odd","docHtml":"\n For every odd number $n$, $φ(2n) = φ(n)$
"},"CarmichaelTotient.charmichaelTotient":{"url":"/FormalConjectures/Wikipedia/CarmichaelTotient/#CarmichaelTotient___charmichaelTotient","anchor":"CarmichaelTotient___charmichaelTotient","docHtml":"\n
\n In Theorem 6 in [F1998], Kevin Ford proves that the smallest counterexample to\nCarmichael's totient function conjecture must be $≥ 10 ^ (10 ^ 10)$
"},"AlmostPerfectNumbers.AlmostPerfect":{"url":"/FormalConjectures/Wikipedia/AlmostPerfectNumbers/#AlmostPerfectNumbers___AlmostPerfect","anchor":"AlmostPerfectNumbers___AlmostPerfect","docHtml":"\n A number is almost perfect if the sum of its divisors is equal to $2n - 1$.
"},"AlmostPerfectNumbers.exists_almost_perfect_not_power_of_two":{"url":"/FormalConjectures/Wikipedia/AlmostPerfectNumbers/#AlmostPerfectNumbers___exists_almost_perfect_not_power_of_two","anchor":"AlmostPerfectNumbers___exists_almost_perfect_not_power_of_two","docHtml":"\nNon-Power-of-2 Almost Perfect Numbers Conjecture.\nDoes there exist an almost perfect number that is not a power of 2?
"},"Hadamard.IsHadamard":{"url":"/FormalConjectures/Wikipedia/Hadamard/#Hadamard___IsHadamard","anchor":"Hadamard___IsHadamard","docHtml":"\n A square matrix $M$ with $±1$-entries that satisfies the equality $|M| ≤ n^\\frac{n}{2}$ is called a
\n Equivalently, a square matrix $M$ with $±1$-entries $|A| ≤ n^\\frac{n}{2}.$ if it satisfies the equality\n$M^TM = n \\cdot 1$, where $1$ denotes the unit matrix.
"},"Hadamard.isHadamard_equiv_isHadamard'":{"url":"/FormalConjectures/Wikipedia/Hadamard/#Hadamard___isHadamard_equiv_isHadamard___","anchor":"Hadamard___isHadamard_equiv_isHadamard___","docHtml":"\n Both definitions are equivalent.
\n\n TOOD(firsching): complete and golf the proof
"},"Hadamard.HadamardConjecture":{"url":"/FormalConjectures/Wikipedia/Hadamard/#Hadamard___HadamardConjecture","anchor":"Hadamard___HadamardConjecture","docHtml":"\n There exists a Hadamard matrix for all $n = 4k$.
"},"Hadamard.exists_hadamard_zero":{"url":"/FormalConjectures/Wikipedia/Hadamard/#Hadamard___exists_hadamard_zero","anchor":"Hadamard___exists_hadamard_zero"},"Hadamard.H12":{"url":"/FormalConjectures/Wikipedia/Hadamard/#Hadamard___H12","anchor":"Hadamard___H12","docHtml":"\n Hadamard constructs a 12 x 12 matrix ...
"},"Hadamard.isHadamard_H12":{"url":"/FormalConjectures/Wikipedia/Hadamard/#Hadamard___isHadamard_H12","anchor":"Hadamard___isHadamard_H12","docHtml":"\n which satisifies the condition.
"},"Hadamard.HadamardConjecture.variants.first_cases":{"url":"/FormalConjectures/Wikipedia/Hadamard/#Hadamard___HadamardConjecture___variants___first_cases","anchor":"Hadamard___HadamardConjecture___variants___first_cases","docHtml":"\n For all $k ≤ 166$, it is known there that there is a Hadamard matrix of size $4 * k$.
"},"Hadamard.HadamardConjecture.variants.«167»":{"url":"/FormalConjectures/Wikipedia/Hadamard/#Hadamard___HadamardConjecture___variants____FLQQ_167_FLQQ_","anchor":"Hadamard___HadamardConjecture___variants____FLQQ_167_FLQQ_","docHtml":"\n The smallest order for which no Hadamard matrix is presently known is $668 = 4 * 167$.
"},"BatemanHornConjecture.OmegaP":{"url":"/FormalConjectures/Wikipedia/BatemanHornConjecture/#BatemanHornConjecture___OmegaP","anchor":"BatemanHornConjecture___OmegaP","docHtml":"\nOmegaP S p counts the number of residue classes mod p where at least one polynomial in S vanishes.
\n The product of degrees of polynomials in a finite set.
"},"BatemanHornConjecture.BatemanHornConstant":{"url":"/FormalConjectures/Wikipedia/BatemanHornConjecture/#BatemanHornConjecture___BatemanHornConstant","anchor":"BatemanHornConjecture___BatemanHornConstant","docHtml":"\n The Bateman-Horn constant of a set of polynomials S. This is defined as the infinite product over all primes:\n$$\\frac{1}{D} \\prod_p (1 - \\frac{1}{p})^{-|S|} (1 - \\frac{\\omega_p(S)}{p})$$\nwhere $D = \\prod_{f \\in S} \\deg(f)$ is the product of degrees and $\\omega_p(S)$ is the number of residue classes mod $p$\nwhere at least one polynomial in $S$ vanishes.
\nCountSimultaneousPrimes S x counts the number of n ≤ x at which all polynomials in S attain a prime value.
\nThe Bateman-Horn Conjecture\nGiven a finite collection of distinct irreducible polynomials non-constant $f_1, f_2, \\dots, f_k \\in \\mathbb{Z}[x]$\nwith positive leading coefficients that satisfy the Schinzel condition, the number\nof positive integers n ≤ x for which all polynomials $f_i$ are simultaneously prime is asymptotic to:\n$$C(f_1, f_2, \\dots, f_k) x / (log x)^k$$\nwhere $C$ is the Bateman-Horn constant given by the convergent infinite product:\n$$C = \\frac{1}{D}\\prod_{p\\in\\mathbb{P}} (1 - 1/p)^(-k) · (1 - \\omega_p/p)$$\nHere $\\omega_p/p$ is the number of residue classes modulo $p$ for which at least one polynomial vanishes.
\n\n The Schinzel condition ensures that for each prime $p$, there exists some integer $n$\nsuch that $p$ does not divide the product $f_(n) f_2(n) \\dotsb f_(n)$, which guarantees the\ninfinite product converges to a positive value.
"},"UnionClosed.IsUnionClosed":{"url":"/FormalConjectures/Wikipedia/UnionClosed/#UnionClosed___IsUnionClosed","anchor":"UnionClosed___IsUnionClosed","docHtml":"\n
\n In this file, we:
\n\n state the conjecture
\n\n state three solved variants of the conjecture, without proof
\n\n prove two solved variants of the conjecture
\n\n prove the conjecture is sharp
\n\n For every finite union-closed family of sets, other than the family containing only the empty set,\nthere exists an element that belongs to at least half of the sets in the family.
"},"UnionClosed.union_closed.variants.yu":{"url":"/FormalConjectures/Wikipedia/UnionClosed/#UnionClosed___union_closed___variants___yu","anchor":"UnionClosed___union_closed___variants___yu","docHtml":"\n Yu [Yu23] showed that the union-closed sets conjecture holds with a constant of approximately\n0.38234 instead of 1/2.\n[Yu23] Yu, Lei (2023). \"Dimension-free bounds for the union-closed sets conjecture\". Entropy. 25 (5): 767.
"},"UnionClosed.union_closed.variants.univ_card":{"url":"/FormalConjectures/Wikipedia/UnionClosed/#UnionClosed___union_closed___variants___univ_card","anchor":"UnionClosed___union_closed___variants___univ_card","docHtml":"\n Vuckovic and Zivkovic [Vu17] showed that the union-closed sets conjecture holds for set families\nwhose universal set has cardinality at most 12.\n[Vu17] Vuckovic, Bojan; Zivkovic, Miodrag (2017). \"The 12-Element Case of Frankl's Conjecture\" (PDF). IPSI BGD Transactions on Internet Research. 13 (1): 65.
"},"UnionClosed.union_closed.variants.family_card":{"url":"/FormalConjectures/Wikipedia/UnionClosed/#UnionClosed___union_closed___variants___family_card","anchor":"UnionClosed___union_closed___variants___family_card","docHtml":"\n Roberts and Simpson [Ro10] showed that the union-closed sets conjecture holds for set families of\nsize at most 46.\nTheir method, however, combined with the result of [Vu17], further shows that it holds for #A ≤ 50\nas well.\n[Ro10] Roberts, Ian; Simpson, Jamie (2010). \"A note on the union-closed sets conjecture\" (PDF). Australas. J. Combin. 47: 265–267.\n[Vu17] Vuckovic, Bojan; Zivkovic, Miodrag (2017). \"The 12-Element Case of Frankl's Conjecture\" (PDF). IPSI BGD Transactions on Internet Research. 13 (1): 65.
\n We can show the union-closed sets conjecture is true for the case where the universal set has\ncardinality 2, by brute force.
"},"UnionClosed.union_closed.variants.singleton_mem":{"url":"/FormalConjectures/Wikipedia/UnionClosed/#UnionClosed___union_closed___variants___singleton_mem","anchor":"UnionClosed___union_closed___variants___singleton_mem","docHtml":"\n We can show the union-closed sets conjecture is true for the case where the set family contains\nsome singleton.
"},"UnionClosed.union_closed.variants.sharpness":{"url":"/FormalConjectures/Wikipedia/UnionClosed/#UnionClosed___union_closed___variants___sharpness","anchor":"UnionClosed___union_closed___variants___sharpness","docHtml":"\n The union-closed sets conjecture is sharp in the sense that if we replace the constant 1/2 with\nany larger constant, then the conjecture fails.
\n If the UC conjecture is tight for some family A then $|A| = 2^k$ for some $k$.
\n Reference: Conjecture 3 in https://www.nieuwarchief.nl/serie5/pdf/naw5-2023-24-4-225.pdf.
"},"Hall.HallIneq":{"url":"/FormalConjectures/Wikipedia/Hall/#Hall___HallIneq","anchor":"Hall___HallIneq","docHtml":"\n There exists a positive number $C$ such that for any integer $x, y$ with $y^2 \\ne x^3$,\n$|y^2 - x^3| > C \\sqrt{|x|}$.
\n\n
\n L. Danilov,
\n Original Hall's conjecture with exponent $1/2$.
"},"Hall.elkies_bound":{"url":"/FormalConjectures/Wikipedia/Hall/#Hall___elkies_bound","anchor":"Hall___elkies_bound","docHtml":"\n Elkies' example $(x, y) = (5853886516781223, 447884928428402042307918)$ shows that such $C$ must be\nless than $0.0215$. Note that simple linarith does not work here.
\n Danilov proved that one cannot replace the exponent $1/2$ with larger number.\nIn other words, for any $\\delta > 0$, there is no positive constant $C$ such that\n$|y^2 - x^3| > C |x| ^ {1/2 + \\delta}$ for all integers $x, y$ with $y^2 \\ne x^3$.
"},"Hall.weak_hall_conjecture":{"url":"/FormalConjectures/Wikipedia/Hall/#Hall___weak_hall_conjecture","anchor":"Hall___weak_hall_conjecture","docHtml":"\n Weak form of Hall's conjecture: relax the exponent from $1/2$ to $1/2 - \\varepsilon$.
"},"GromovPolynomialGrowth.growthFunction_not_polynomial_of_infinite":{"url":"/FormalConjectures/Wikipedia/GromovPolynomialGrowth/#GromovPolynomialGrowth___growthFunction_not_polynomial_of_infinite","anchor":"GromovPolynomialGrowth___growthFunction_not_polynomial_of_infinite","docHtml":"\n Infinite groups do not satisfy polynomial growth over ℕ for any degree d because when\nd = 0 this reduces to the unbounded nature of growthFunction while n = 0 works when d ≠ 0.\nThus a finitely-generated infinite nilpotent group would be a counter-example to\nGromov's theorem when quantifying over all of ℕ, and so n = 0 should be excluded.
\nGromov's Polynomial Growth Theorem : A finitely generated group has\npolynomial growth if and only if it is virtually nilpotent.
"},"Transcendental.exp_add_pi_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___exp_add_pi_transcendental","anchor":"Transcendental___exp_add_pi_transcendental","docHtml":"\n $e + \\pi$ is transcendental.
"},"Transcendental.exp_mul_pi_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___exp_mul_pi_transcendental","anchor":"Transcendental___exp_mul_pi_transcendental","docHtml":"\n $e\\pi$ is transcendental.
"},"Transcendental.exp_pow_pi_sq_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___exp_pow_pi_sq_transcendental","anchor":"Transcendental___exp_pow_pi_sq_transcendental","docHtml":"\n $e^{\\pi^2}$ is transcendental.
"},"Transcendental.exp_exp_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___exp_exp_transcendental","anchor":"Transcendental___exp_exp_transcendental","docHtml":"\n $e^e$ is transcendental.
"},"Transcendental.pi_pow_exp_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___pi_pow_exp_transcendental","anchor":"Transcendental___pi_pow_exp_transcendental","docHtml":"\n $\\pi^e$ is transcendental.
"},"Transcendental.pi_pow_sqrt_two_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___pi_pow_sqrt_two_transcendental","anchor":"Transcendental___pi_pow_sqrt_two_transcendental","docHtml":"\n $\\pi^{\\sqrt{2}}$ is transcendental.
"},"Transcendental.pi_pow_pi_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___pi_pow_pi_transcendental","anchor":"Transcendental___pi_pow_pi_transcendental","docHtml":"\n $\\pi^{\\pi}$ is transcendental.
"},"Transcendental.pi_pow_pi_pow_pi_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___pi_pow_pi_pow_pi_transcendental","anchor":"Transcendental___pi_pow_pi_pow_pi_transcendental","docHtml":"\n $\\pi^{\\pi^{\\pi}}$ is transcendental.
"},"Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___pi_pow_pi_pow_pi_pow_pi_transcendental","anchor":"Transcendental___pi_pow_pi_pow_pi_pow_pi_transcendental","docHtml":"\n $\\pi^{\\pi^{\\pi^\\pi}}$ is transcendental.
"},"Transcendental.pi_pow_pi_pow_pi_pow_pi_not_integer":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___pi_pow_pi_pow_pi_pow_pi_not_integer","anchor":"Transcendental___pi_pow_pi_pow_pi_pow_pi_not_integer","docHtml":"\n $\\pi^{\\pi^{\\pi^\\pi}}$ is not an integer.
\n\n This would follow from $\\pi^{\\pi^{\\pi^\\pi}}$ being transcendental,\nbut this formulation is of interest in its own right,\nas it could in principle be proven by direct computation.
\n\n
\n $\\log(\\pi)$ is transcendental.
"},"Transcendental.rlog_rlog_two_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___rlog_rlog_two_transcendental","anchor":"Transcendental___rlog_rlog_two_transcendental","docHtml":"\n $\\log(\\log(2))$ is transcendental.
"},"Transcendental.sin_exp_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___sin_exp_transcendental","anchor":"Transcendental___sin_exp_transcendental","docHtml":"\n $\\sin(e)$ is transcendental.
"},"Transcendental.exp_add_pi_or_exp_add_mul_transcendental":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___exp_add_pi_or_exp_add_mul_transcendental","anchor":"Transcendental___exp_add_pi_or_exp_add_mul_transcendental","docHtml":"\n At least one of $\\pi + e$ and $\\pi e$ is transcendental.
"},"Transcendental.transcendental_catalanConstant_or_gompertzConstant":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___transcendental_catalanConstant_or_gompertzConstant","anchor":"Transcendental___transcendental_catalanConstant_or_gompertzConstant","docHtml":"\n At least one of Catalan constant and the Gompertz constant is transcendental.
"},"Transcendental.transcendental_catalanConstant":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___transcendental_catalanConstant","anchor":"Transcendental___transcendental_catalanConstant","docHtml":"\n The Catalan constant $G$ is transcendental.
"},"Transcendental.transcendental_gompertzConstant":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___transcendental_gompertzConstant","anchor":"Transcendental___transcendental_gompertzConstant","docHtml":"\n The Gompertz constant $\\delta$ is transcendental.
"},"Transcendental.transcendental_gamma_one_div_two":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___transcendental_gamma_one_div_two","anchor":"Transcendental___transcendental_gamma_one_div_two","docHtml":"\n $\\Gamma(1/2)$ is transcendental.
\n\n [Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.
"},"Transcendental.transcendental_gamma_one_div_three":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___transcendental_gamma_one_div_three","anchor":"Transcendental___transcendental_gamma_one_div_three","docHtml":"\n $\\Gamma(1/3)$ is transcendental.
\n\n [Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.
"},"Transcendental.transcendental_gamma_one_div_four":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___transcendental_gamma_one_div_four","anchor":"Transcendental___transcendental_gamma_one_div_four","docHtml":"\n $\\Gamma(1/4)$ is transcendental.
\n\n [Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.
"},"Transcendental.transcendental_gamma_one_div_six":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___transcendental_gamma_one_div_six","anchor":"Transcendental___transcendental_gamma_one_div_six","docHtml":"\n $\\Gamma(1/6)$ is transcendental.
\n\n [Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.
"},"Transcendental.transcendental_gamma_one_div":{"url":"/FormalConjectures/Wikipedia/Transcendental/#Transcendental___transcendental_gamma_one_div","anchor":"Transcendental___transcendental_gamma_one_div","docHtml":"\n $\\Gamma(1/n)$ for n ≥ 2 is transcendental.
\n
\n The Beal Conjecture: if we are given positive integers $A, B, C, x, y, z$ such that\n$x, y, z > 2$ and $A^x + B^y = C^z$ then $A, B, C$ have a common divisor.
"},"BealConjecture.flt_of_beal_conjecture":{"url":"/FormalConjectures/Wikipedia/BealConjecture/#BealConjecture___flt_of_beal_conjecture","anchor":"BealConjecture___flt_of_beal_conjecture","docHtml":"\n The Beal Conjecture implies Fermat's last theorem
"},"LychrelNumbers.base":{"url":"/FormalConjectures/Wikipedia/LychrelNumbers/#LychrelNumbers___base","anchor":"LychrelNumbers___base","docHtml":"\n The base (10) used for digit reversal.
"},"LychrelNumbers.rev10":{"url":"/FormalConjectures/Wikipedia/LychrelNumbers/#LychrelNumbers___rev10","anchor":"LychrelNumbers___rev10","docHtml":"\n The digit-reversal map $\\operatorname{rev}_{10}(n)$.
\n\n Implementation note: Nat.digits base n returns the digits of n in
\n A number is a (base-10) palindrome if it equals its digit reversal.
"},"LychrelNumbers.lychrelStep":{"url":"/FormalConjectures/Wikipedia/LychrelNumbers/#LychrelNumbers___lychrelStep","anchor":"LychrelNumbers___lychrelStep","docHtml":"\n One step of the Lychrel iteration: n ↦ n + rev10 n.
\n The number $n$ is a (base-10) Lychrel number if no iterate of the Lychrel process is a palindrome.
"},"LychrelNumbers.no_lychrel_numbers_base10":{"url":"/FormalConjectures/Wikipedia/LychrelNumbers/#LychrelNumbers___no_lychrel_numbers_base10","anchor":"LychrelNumbers___no_lychrel_numbers_base10","docHtml":"\nLychrel conjecture (base 10): conjecturally, there are no Lychrel numbers in base 10.
\n\n Equivalently, every positive integer eventually becomes a palindrome under the Lychrel iteration.
"},"LychrelNumbers.isLychrel10_196":{"url":"/FormalConjectures/Wikipedia/LychrelNumbers/#LychrelNumbers___isLychrel10_196","anchor":"LychrelNumbers___isLychrel10_196","docHtml":"\n The first widely studied open case: whether 196 is a base-10 Lychrel number.
\n An equivalent formulation of no_lychrel_numbers_base10.
\n Sanity check: digit reversal of 120 is 21.
\n Sanity check: 121 is a base-10 palindrome.
\n Sanity check: 56 → 121 in one Lychrel step.
\n Sanity check: the Lychrel iteration at 56 reaches a palindrome.
\n The horizontal side of the hallway is $(-\\infty, 1] \\times [0, 1]$.
"},"MovingSofa.verticalHallway":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___verticalHallway","anchor":"MovingSofa___verticalHallway","docHtml":"\n The vertical side of the hallway is $[0, 1] \\times (-\\infty, 1]$.
"},"MovingSofa.hallway":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___hallway","anchor":"MovingSofa___hallway","docHtml":"\n The hallway is the union of its horizontal and vertical sides.
"},"MovingSofa.IsMovingSofa":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___IsMovingSofa","anchor":"MovingSofa___IsMovingSofa","docHtml":"\n A connected closed set $s$ is a moving sofa according to a rigid motion $m:I\\to\\mathrm{SE}(2)$,\nif the sofa is initially in the horizontal side of the hallway and ends up in the vertical side.\nHere, since $\\mathrm{SE}(2)$ is not in Mathlib yet, we use $\\mathrm{E}(2)$ and rely on continuity\nand $m(0) = \\mathrm{id}$ to ensure $m$ is in $\\mathrm{SE}(2)$.
"},"MovingSofa.IsMovingSofa.isConnected":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___IsMovingSofa___isConnected","anchor":"MovingSofa___IsMovingSofa___isConnected","docHtml":"\n A connected closed set $s$ is a moving sofa according to a rigid motion $m:I\\to\\mathrm{SE}(2)$,\nif the sofa is initially in the horizontal side of the hallway and ends up in the vertical side.\nHere, since $\\mathrm{SE}(2)$ is not in Mathlib yet, we use $\\mathrm{E}(2)$ and rely on continuity\nand $m(0) = \\mathrm{id}$ to ensure $m$ is in $\\mathrm{SE}(2)$.
"},"MovingSofa.IsMovingSofa.isClosed":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___IsMovingSofa___isClosed","anchor":"MovingSofa___IsMovingSofa___isClosed","docHtml":"\n A connected closed set $s$ is a moving sofa according to a rigid motion $m:I\\to\\mathrm{SE}(2)$,\nif the sofa is initially in the horizontal side of the hallway and ends up in the vertical side.\nHere, since $\\mathrm{SE}(2)$ is not in Mathlib yet, we use $\\mathrm{E}(2)$ and rely on continuity\nand $m(0) = \\mathrm{id}$ to ensure $m$ is in $\\mathrm{SE}(2)$.
"},"MovingSofa.IsMovingSofa.continuous":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___IsMovingSofa___continuous","anchor":"MovingSofa___IsMovingSofa___continuous","docHtml":"\n A connected closed set $s$ is a moving sofa according to a rigid motion $m:I\\to\\mathrm{SE}(2)$,\nif the sofa is initially in the horizontal side of the hallway and ends up in the vertical side.\nHere, since $\\mathrm{SE}(2)$ is not in Mathlib yet, we use $\\mathrm{E}(2)$ and rely on continuity\nand $m(0) = \\mathrm{id}$ to ensure $m$ is in $\\mathrm{SE}(2)$.
"},"MovingSofa.IsMovingSofa.zero":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___IsMovingSofa___zero","anchor":"MovingSofa___IsMovingSofa___zero","docHtml":"\n A connected closed set $s$ is a moving sofa according to a rigid motion $m:I\\to\\mathrm{SE}(2)$,\nif the sofa is initially in the horizontal side of the hallway and ends up in the vertical side.\nHere, since $\\mathrm{SE}(2)$ is not in Mathlib yet, we use $\\mathrm{E}(2)$ and rely on continuity\nand $m(0) = \\mathrm{id}$ to ensure $m$ is in $\\mathrm{SE}(2)$.
"},"MovingSofa.IsMovingSofa.initial":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___IsMovingSofa___initial","anchor":"MovingSofa___IsMovingSofa___initial","docHtml":"\n A connected closed set $s$ is a moving sofa according to a rigid motion $m:I\\to\\mathrm{SE}(2)$,\nif the sofa is initially in the horizontal side of the hallway and ends up in the vertical side.\nHere, since $\\mathrm{SE}(2)$ is not in Mathlib yet, we use $\\mathrm{E}(2)$ and rely on continuity\nand $m(0) = \\mathrm{id}$ to ensure $m$ is in $\\mathrm{SE}(2)$.
"},"MovingSofa.IsMovingSofa.subset_hallway":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___IsMovingSofa___subset_hallway","anchor":"MovingSofa___IsMovingSofa___subset_hallway","docHtml":"\n A connected closed set $s$ is a moving sofa according to a rigid motion $m:I\\to\\mathrm{SE}(2)$,\nif the sofa is initially in the horizontal side of the hallway and ends up in the vertical side.\nHere, since $\\mathrm{SE}(2)$ is not in Mathlib yet, we use $\\mathrm{E}(2)$ and rely on continuity\nand $m(0) = \\mathrm{id}$ to ensure $m$ is in $\\mathrm{SE}(2)$.
"},"MovingSofa.IsMovingSofa.final":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___IsMovingSofa___final","anchor":"MovingSofa___IsMovingSofa___final","docHtml":"\n A connected closed set $s$ is a moving sofa according to a rigid motion $m:I\\to\\mathrm{SE}(2)$,\nif the sofa is initially in the horizontal side of the hallway and ends up in the vertical side.\nHere, since $\\mathrm{SE}(2)$ is not in Mathlib yet, we use $\\mathrm{E}(2)$ and rely on continuity\nand $m(0) = \\mathrm{id}$ to ensure $m$ is in $\\mathrm{SE}(2)$.
"},"MovingSofa.unitSquare":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___unitSquare","anchor":"MovingSofa___unitSquare","docHtml":"\n The unit square.
"},"_private.0.MovingSofa.mem_Icc_of_mem_unitSquare":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#_private___0___MovingSofa___mem_Icc_of_mem_unitSquare","anchor":"_private___0___MovingSofa___mem_Icc_of_mem_unitSquare","docHtml":"\n Coordinates of points in the unit square lie in [0,1].
\n The unit square $[0,1]^2$ is a valid moving sofa (with the identity motion).\nIt sits in the corner where both hallways overlap, so the stationary motion works.\nThis is a sanity check that the IsMovingSofa definition is not vacuous.
\n The rigid motion that translates by $p$ and then rotates counterclockwise by $\\alpha$.\nNote that [Ge92] used this definition while [Ro18] used rotation first and then translation.
"},"MovingSofa.sofaOfRotateTranslatePath":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___sofaOfRotateTranslatePath","anchor":"MovingSofa___sofaOfRotateTranslatePath","docHtml":"\n The sofa according to a rotation path $p : [0, \\pi/2] \\to \\mathbb{R}^2$ as in [Ge92] is the\nintersection over $\\alpha \\in [0, \\pi/2]$ of hallways each translated by $p(\\alpha)$ and then\nrotated by $\\alpha$, with the special cases that the hallway at $0$ is the horizontal side\nand the hallway at $\\pi/2$ is the vertical side.
"},"MovingSofa.GerversSofa.ABφθSpec":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___GerversSofa___AB______Spec","anchor":"MovingSofa___GerversSofa___AB______Spec","docHtml":"\n Eq. 1-4 of [Ro18], which specifies the constants $A$, $B$, $\\varphi$, and $\\theta$ of [Ge92].
"},"MovingSofa.GerversSofa.ABφθSpec.existsUnique":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___GerversSofa___AB______Spec___existsUnique","anchor":"MovingSofa___GerversSofa___AB______Spec___existsUnique","docHtml":"\n There exist unique constants $A$, $B$, $\\varphi$, and $\\theta$ satisfying the spec.
"},"MovingSofa.GerversSofa.A":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___GerversSofa___A","anchor":"MovingSofa___GerversSofa___A"},"MovingSofa.GerversSofa.B":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___GerversSofa___B","anchor":"MovingSofa___GerversSofa___B"},"MovingSofa.GerversSofa.φ":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___GerversSofa______","anchor":"MovingSofa___GerversSofa______"},"MovingSofa.GerversSofa.θ":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___GerversSofa______-","anchor":"MovingSofa___GerversSofa______-"},"MovingSofa.GerversSofa.r":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___GerversSofa___r","anchor":"MovingSofa___GerversSofa___r"},"MovingSofa.GerversSofa.y":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___GerversSofa___y","anchor":"MovingSofa___GerversSofa___y"},"MovingSofa.GerversSofa.x":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___GerversSofa___x","anchor":"MovingSofa___GerversSofa___x"},"MovingSofa.GerversSofa.p":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___GerversSofa___p","anchor":"MovingSofa___GerversSofa___p"},"MovingSofa.gerversSofa":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___gerversSofa","anchor":"MovingSofa___gerversSofa","docHtml":"\n Gerver's sofa is the sofa according to the rotation path GerversSofa.p.
\n The sofa constant is the maximal area of a moving sofa.
"},"MovingSofa.one_le_sofaConstant":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___one_le_sofaConstant","anchor":"MovingSofa___one_le_sofaConstant","docHtml":"\n The sofa constant is at least 1, as witnessed by the unit square.
"},"MovingSofa.sofaConstant_eq":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___sofaConstant_eq","anchor":"MovingSofa___sofaConstant_eq","docHtml":"\n What is the sofa constant?
"},"MovingSofa.sofaConstant_eq_volume_gerversSofa":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___sofaConstant_eq_volume_gerversSofa","anchor":"MovingSofa___sofaConstant_eq_volume_gerversSofa","docHtml":"\n Gerver's sofa attains the sofa constant, conjectured by [Ge92] and claimed by [Ba24].
"},"MovingSofa.sofaConstant_eq_volume_iff_eq_gerversSofa":{"url":"/FormalConjectures/Wikipedia/MovingSofa/#MovingSofa___sofaConstant_eq_volume_iff_eq_gerversSofa","anchor":"MovingSofa___sofaConstant_eq_volume_iff_eq_gerversSofa","docHtml":"\n Gerver's sofa is the unique sofa that attains the sofa constant.
"},"Exponentials.four_exponentials_conjecture":{"url":"/FormalConjectures/Wikipedia/Exponentials/#Exponentials___four_exponentials_conjecture","anchor":"Exponentials___four_exponentials_conjecture","docHtml":"\nFour exponentials conjecture\nLet $x_0, x_1$ and $y_0, y_1$ be $\\mathbb Q$-linearly independent pairs of complex numbers,\nthen some $e^{x_i y_j}$ is transcendental.
"},"Exponentials.two_pow_three_pow_transcendental":{"url":"/FormalConjectures/Wikipedia/Exponentials/#Exponentials___two_pow_three_pow_transcendental","anchor":"Exponentials___two_pow_three_pow_transcendental","docHtml":"\n The four exponential conjecture would imply that for any irrational number $t$,\nat least one of the numbers $2^t$ and $3^t$ is transcendental.
"},"SierpinskiNumber.selfridge_78557":{"url":"/FormalConjectures/Wikipedia/SierpinskiNumber/#SierpinskiNumber___selfridge_78557","anchor":"SierpinskiNumber___selfridge_78557","docHtml":"\n Selfridge proved in 1962 that 78557 is a Sierpiński number by showing that all numbers of the\nform $78557 \\cdot 2^n + 1$ have a factor in the covering set ${3, 5, 7, 13, 19, 37, 73}$.
"},"SierpinskiNumber.selfridge_conjecture":{"url":"/FormalConjectures/Wikipedia/SierpinskiNumber/#SierpinskiNumber___selfridge_conjecture","anchor":"SierpinskiNumber___selfridge_conjecture","docHtml":"\nThe Sierpiński problem (Selfridge's conjecture). Is 78557 the smallest Sierpiński number?
\n\n Selfridge conjectured that 78557 is the smallest Sierpiński number. He proved in 1962 that\n78557 is indeed a Sierpiński number by showing that all numbers of the form $78557 \\cdot 2^n + 1$\nhave a factor in the covering set ${3, 5, 7, 13, 19, 37, 73}$.
"},"SierpinskiNumber.prime_sierpinski_problem":{"url":"/FormalConjectures/Wikipedia/SierpinskiNumber/#SierpinskiNumber___prime_sierpinski_problem","anchor":"SierpinskiNumber___prime_sierpinski_problem","docHtml":"\nThe prime Sierpiński problem. Is 271129 the smallest prime Sierpiński number?
\n\n In 1976, Nathan Mendelsohn determined that the second provable Sierpiński number is the prime\n$k = 271129$.
"},"SierpinskiNumber.extended_sierpinski_problem":{"url":"/FormalConjectures/Wikipedia/SierpinskiNumber/#SierpinskiNumber___extended_sierpinski_problem","anchor":"SierpinskiNumber___extended_sierpinski_problem","docHtml":"\nThe extended Sierpiński problem. Is 271129 the second-smallest Sierpiński number?
\n\n Even if 78557 is confirmed as the smallest Sierpiński number, there could exist a composite\nSierpiński number $k$ with $78557 < k < 271129$. We formalize \"second-smallest\" as: the\nleast Sierpiński number $k$ such that there exists exactly one Sierpiński number below it.
"},"CongruentNumber.congruentNumber":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___congruentNumber","anchor":"CongruentNumber___congruentNumber","docHtml":"\n A natural number $n$ is called a congruent number if there exists a right triangle with rational\nsides $a$, $b$, and hypotenuse $c$ such that the area of the triangle is $\\frac{1}{2}ab = n$.
\n\n
\n 1 is not a congruent number, as proved by Fermat via infinite descent.
"},"CongruentNumber.congruentNumber_5":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___congruentNumber_5","anchor":"CongruentNumber___congruentNumber_5"},"CongruentNumber.congruentNumber_6":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___congruentNumber_6","anchor":"CongruentNumber___congruentNumber_6"},"CongruentNumber.congruentNumber_7":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___congruentNumber_7","anchor":"CongruentNumber___congruentNumber_7"},"CongruentNumber.congruentNumber_157_zagier":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___congruentNumber_157_zagier","anchor":"CongruentNumber___congruentNumber_157_zagier"},"CongruentNumber.A":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___A","anchor":"CongruentNumber___A"},"CongruentNumber.B":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___B","anchor":"CongruentNumber___B"},"CongruentNumber.C":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___C","anchor":"CongruentNumber___C"},"CongruentNumber.D":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___D","anchor":"CongruentNumber___D"},"CongruentNumber.Tunnell_odd":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___Tunnell_odd","anchor":"CongruentNumber___Tunnell_odd","docHtml":"\n Tunnell's theorem (necessary condition) for odd squarefree congruent numbers.
"},"CongruentNumber.Tunnell_even":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___Tunnell_even","anchor":"CongruentNumber___Tunnell_even","docHtml":"\n Tunnell's theorem (necessary condition) for even squarefree congruent numbers.
"},"CongruentNumber.Tunnell_odd_converse":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___Tunnell_odd_converse","anchor":"CongruentNumber___Tunnell_odd_converse","docHtml":"\n Tunnell's theorem (sufficient condition assuming BSD) for odd squarefree congruent numbers.
"},"CongruentNumber.Tunnell_even_converse":{"url":"/FormalConjectures/Wikipedia/CongruentNumber/#CongruentNumber___Tunnell_even_converse","anchor":"CongruentNumber___Tunnell_even_converse","docHtml":"\n Tunnell's theorem (sufficient condition assuming BSD) for even squarefree congruent numbers.
"},"Conjecture_1_3_to_2_3.conjecture_1_3_to_2_3":{"url":"/FormalConjectures/Wikipedia/conjecture_1_3_to_2_3/#Conjecture_1_3_to_2_3___conjecture_1_3_to_2_3","anchor":"Conjecture_1_3_to_2_3___conjecture_1_3_to_2_3","docHtml":"\n Does every finite partially ordered set that is not totally ordered\ncontain two elements $x$ and $y$ such that the probability that\n$x$ appears before $y$ in a random linear extension is between $\\frac 1 3$ and $\\frac 2 3$?
\n\n The set of all total order extensions is represented as order preserving\nbijections $P$ of $1, ..., n$.
"},"Oppermann.oppermann_conjecture.parts.i":{"url":"/FormalConjectures/Wikipedia/Oppermann/#Oppermann___oppermann_conjecture___parts___i","anchor":"Oppermann___oppermann_conjecture___parts___i","docHtml":"\n For every integer $x \\ge 2$ there exists a prime between $x(x-1)$ and $x^2$.
"},"Oppermann.oppermann_conjecture.parts.ii":{"url":"/FormalConjectures/Wikipedia/Oppermann/#Oppermann___oppermann_conjecture___parts___ii","anchor":"Oppermann___oppermann_conjecture___parts___ii","docHtml":"\n For every integer $x \\ge 2$ there exists a prime between $x^2$ and $x(x+1)$.
"},"Oppermann.oppermann_conjecture":{"url":"/FormalConjectures/Wikipedia/Oppermann/#Oppermann___oppermann_conjecture","anchor":"Oppermann___oppermann_conjecture","docHtml":"\nOppermann's Conjecture:\nFor every integer $x \\ge 2$, the following hold:
\n\n There exists a prime between $x(x-1)$ and $x^2$.
\n\n There exists a prime between $x^2$ and $x(x+1)$.
\n\n Oppermann's conjecture implies Brocard's conjecture.
"},"Oppermann.oppermann_implies_legendre":{"url":"/FormalConjectures/Wikipedia/Oppermann/#Oppermann___oppermann_implies_legendre","anchor":"Oppermann___oppermann_implies_legendre","docHtml":"\n Oppermann's conjecture implies Legendre's conjecture.
"},"Oppermann.oppermann_conjecture.ferreira_large_x":{"url":"/FormalConjectures/Wikipedia/Oppermann/#Oppermann___oppermann_conjecture___ferreira_large_x","anchor":"Oppermann___oppermann_conjecture___ferreira_large_x","docHtml":"\n Ferreira proved that Oppermann's conjecture is true for sufficiently large x.
"},"Schinzel.schinzel_conjecture":{"url":"/FormalConjectures/Wikipedia/Schinzel/#Schinzel___schinzel_conjecture","anchor":"Schinzel___schinzel_conjecture","docHtml":"\nSchinzel conjecture (H hypothesis)\nIf a finite set of polynomials $f_i$ satisfies both Schinzel and Bunyakovsky conditions,\nthere exist infinitely many natural numbers $n$ such that $f_i(n)$ are primes for all $i$.
"},"DeterminantalConjecture.determinantal_conjecture":{"url":"/FormalConjectures/Wikipedia/DeterminantalConjecture/#DeterminantalConjecture___determinantal_conjecture","anchor":"DeterminantalConjecture___determinantal_conjecture","docHtml":"\n Does the determinant of the sum $A + B$ of two $n \\times n$ normal\ncomplex matrices $A$ and $B$ always lie in the convex hull\nof the $n!$ points $\\prod_i (\\lambda(A)
\n There are two conjectures related to the Ramanujan τ-function:
\n\n Ramanujan-Petersson conjecture: For every prime p, the absolute value of the\nRamanujan τ-function at p is bounded by 2 * p^(11/2).
\n Lehmer's conjecture: The Ramanujan τ-function is never zero for any positive integer n.
\n
\n The Ramanujan-Petersson conjecture: $|\\tau(p)| \\le 2 p^{11/2}$ for primes $p$.
"},"RamanujanTau.lehmer_ramanujan_tau":{"url":"/FormalConjectures/Wikipedia/RamanujanTau/#RamanujanTau___lehmer_ramanujan_tau","anchor":"RamanujanTau___lehmer_ramanujan_tau","docHtml":"\n Lehmer's conjecture: $\\tau(n) \\ne 0$ for all $n > 0$.
"},"RiemannZetaValues.irrational_five":{"url":"/FormalConjectures/Wikipedia/RiemannZetaValues/#RiemannZetaValues___irrational_five","anchor":"RiemannZetaValues___irrational_five","docHtml":"\n $\\zeta(5)$ is irrational.
"},"RiemannZetaValues.irrational_seven":{"url":"/FormalConjectures/Wikipedia/RiemannZetaValues/#RiemannZetaValues___irrational_seven","anchor":"RiemannZetaValues___irrational_seven","docHtml":"\n $\\zeta(7)$ is irrational.
"},"RiemannZetaValues.irrational_nine":{"url":"/FormalConjectures/Wikipedia/RiemannZetaValues/#RiemannZetaValues___irrational_nine","anchor":"RiemannZetaValues___irrational_nine","docHtml":"\n $\\zeta(9)$ is irrational.
"},"RiemannZetaValues.irrational_eleven":{"url":"/FormalConjectures/Wikipedia/RiemannZetaValues/#RiemannZetaValues___irrational_eleven","anchor":"RiemannZetaValues___irrational_eleven","docHtml":"\n $\\zeta(11)$ is irrational.
"},"RiemannZetaValues.irrational_odd":{"url":"/FormalConjectures/Wikipedia/RiemannZetaValues/#RiemannZetaValues___irrational_odd","anchor":"RiemannZetaValues___irrational_odd","docHtml":"\n $\\zeta(2n + 1)$ is irrational for any $n\\in\\mathbb{N}^{+}$.
"},"RiemannZetaValues.irrational_three":{"url":"/FormalConjectures/Wikipedia/RiemannZetaValues/#RiemannZetaValues___irrational_three","anchor":"RiemannZetaValues___irrational_three","docHtml":"\n $\\zeta(3)$ is irrational.
\n\n [Ap79] Apéry, R. (1979).
\n There are infinitely many $\\zeta(2n + 1)$, $n \\in \\mathbb{N}$, that are irrational.
\n\n [Ri00] Rivoal, T. (2000).
\n At least one of $\\zeta(5), \\zeta(7), \\zeta(9)$ or $\\zeta(11)$ is irrational.
\n\n [Zu01] W. Zudilin (2001).
\n The Agoh-Giuga Conjecture, Agoh's formulation.\nAn integer p ≥ 2 is prime if and only if we have\np*B_{p-1} ≡ -1 [MOD p]
\n The Agoh-Giuga Conjecture, Giuga's formulation.\nAn integer p ≥ 2 is prime if and only if it satifies the congruence\n∑_{i=1}^{p-1} i^{p-1} ≡ -1 [MOD p].
\n The Agoh-Giuga Conjecture, Agoh's formulation
"},"AgohGiuga.agoh_giuga.variants.giuga":{"url":"/FormalConjectures/Wikipedia/AgohGiuga/#AgohGiuga___agoh_giuga___variants___giuga","anchor":"AgohGiuga___agoh_giuga___variants___giuga","docHtml":"\n The Agoh-Giuga Conjecture, Giuga's formulation
"},"AgohGiuga.agoh_giuga.variants.equivalence":{"url":"/FormalConjectures/Wikipedia/AgohGiuga/#AgohGiuga___agoh_giuga___variants___equivalence","anchor":"AgohGiuga___agoh_giuga___variants___equivalence","docHtml":"\n The two statements of the conjecture are equivalent.
"},"AgohGiuga.IsWeakGiuga":{"url":"/FormalConjectures/Wikipedia/AgohGiuga/#AgohGiuga___IsWeakGiuga","anchor":"AgohGiuga___IsWeakGiuga","docHtml":"\n A (weak) Giuga number is a composite number $n$ such that\n$$\\sum_{i=1}^{n - 1}i^{\\varphi(n)} \\equiv -1\\pmod{n}$$.
"},"AgohGiuga.IsStrongGiuga":{"url":"/FormalConjectures/Wikipedia/AgohGiuga/#AgohGiuga___IsStrongGiuga","anchor":"AgohGiuga___IsStrongGiuga","docHtml":"\n A (strong) Giuga number is a composite number $n$ such that\n$$\\sum_{i=1}^{n - 1}i^{n - 1} \\equiv -1\\pmod{n}$$
"},"AgohGiuga.isWeakGiuga_iff_prime_dvd":{"url":"/FormalConjectures/Wikipedia/AgohGiuga/#AgohGiuga___isWeakGiuga_iff_prime_dvd","anchor":"AgohGiuga___isWeakGiuga_iff_prime_dvd","docHtml":"\n A composite number $n$ is weak Giuga if and only if $p \\mid (\\frac{n}{p} - 1)$ for all\nprime divisors $p$ of $n$.
"},"AgohGiuga.isWeakGiuga_iff_sum_primeFactors":{"url":"/FormalConjectures/Wikipedia/AgohGiuga/#AgohGiuga___isWeakGiuga_iff_sum_primeFactors","anchor":"AgohGiuga___isWeakGiuga_iff_sum_primeFactors","docHtml":"\n A composite number $n$ is weak Giuga if and only if\n$$\n\\sum_{p\\mid n} \\frac{1}{p} - \\frac{1}{n} \\in\\mathbb{N}.\n$$
"},"AgohGiuga.IsCarmichael":{"url":"/FormalConjectures/Wikipedia/AgohGiuga/#AgohGiuga___IsCarmichael","anchor":"AgohGiuga___IsCarmichael","docHtml":"\n A Carmichael number is a composite number n such that for all b ≥ 1,\nwe have b^n ≡ b (mod n).
\n A composite Carmichael number is squarefree.
"},"AgohGiuga.korselts_criterion":{"url":"/FormalConjectures/Wikipedia/AgohGiuga/#AgohGiuga___korselts_criterion","anchor":"AgohGiuga___korselts_criterion","docHtml":"\n A composite number a is Carmichael if and only if it is squarefree\nand, for all prime p dividing a, we have p - 1 ∣ a - 1.
\n Giuga showed that a number n is strong Giuga if and only if it is\nCarmichael and ∑_{p|n} 1/p - 1/n ∈ ℕ (i.e., if and only if it is Carmichael\nand weak Giuga).\nRef: G. Giuga,
\n Every strong Giuga number is a Carmichael number.
"},"AgohGiuga.agoh_giuga.variants.le_primeFactors_card_of_isStrongGiuga":{"url":"/FormalConjectures/Wikipedia/AgohGiuga/#AgohGiuga___agoh_giuga___variants___le_primeFactors_card_of_isStrongGiuga","anchor":"AgohGiuga___agoh_giuga___variants___le_primeFactors_card_of_isStrongGiuga","docHtml":"\n Giuga showed that a Giuga number has at least 9 prime factors.\nRef: G. Giuga,
\n Giuga showed that a counterexample Giuga number has at least 1000 digits.\nRef: G. Giuga,
\n Bedocchi showed that any Giuga number has at least 1700 digits.\nRef: E. Bedocchi,
\n Borwein, Borwein, Borwein and Girgensohn showed that any strong Giuga\nnumber has at least 13000 digits.\nRef: D. Borwein, J. M. Borwein, P. B. Borwein, and R. Girgensohn,
\n Let G(X) denote the number of exceptions n ≤ X to Giuga’s conjecture.\nThen for X larger than an absolute constant which can be made\nexplicit, G(X) ≪ X^{1/2} log X.\nRef: Vicentiu Tipu,
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\n The Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.
\n\n
\nBB(n) is the n-th Busy Beaver number.\n
\n To compute BB n, we need only consider machines with states and symbols indexed in Fin.
\n The value of the Busy Beaver function for 1 state is 1.
"},"BusyBeaver.BB_2":{"url":"/FormalConjectures/Wikipedia/BusyBeaver/#BusyBeaver___BB_2","anchor":"BusyBeaver___BB_2","docHtml":"\n The value of the Busy Beaver function for 2 states is 6.
"},"BusyBeaver.BB_3":{"url":"/FormalConjectures/Wikipedia/BusyBeaver/#BusyBeaver___BB_3","anchor":"BusyBeaver___BB_3","docHtml":"\n The value of the Busy Beaver function for 3 states is 21.
"},"BusyBeaver.BB_4":{"url":"/FormalConjectures/Wikipedia/BusyBeaver/#BusyBeaver___BB_4","anchor":"BusyBeaver___BB_4","docHtml":"\n The value of the Busy Beaver function for 4 states is 107.
"},"BusyBeaver.BB_5":{"url":"/FormalConjectures/Wikipedia/BusyBeaver/#BusyBeaver___BB_5","anchor":"BusyBeaver___BB_5","docHtml":"\n The value of the Busy Beaver function for 5 states is 47176870.
"},"BusyBeaver.BB_6":{"url":"/FormalConjectures/Wikipedia/BusyBeaver/#BusyBeaver___BB_6","anchor":"BusyBeaver___BB_6","docHtml":"\n Determine the value of the Busy Beaver function at n = 6.
"},"WolstenholmePrime.wolstenholme_theorem":{"url":"/FormalConjectures/Wikipedia/WolstenholmePrime/#WolstenholmePrime___wolstenholme_theorem","anchor":"WolstenholmePrime___wolstenholme_theorem","docHtml":"\n Wolstenholme's theorem states that any prime $p > 3$ satisfies $\\binom{2p-1}{p-1} \\equiv 1 (\\pmod{p^3})$.
\n\n Formal proof linked here provided by AlphaProof.\n
\n A prime $p > 7$ is called a
\n Two known Wolstenholme primes: 16843 and 2124679.
\n\n Formal proof linked here provided by AlphaProof
"},"WolstenholmePrime.wolstenholme_prime_2124679":{"url":"/FormalConjectures/Wikipedia/WolstenholmePrime/#WolstenholmePrime___wolstenholme_prime_2124679","anchor":"WolstenholmePrime___wolstenholme_prime_2124679"},"WolstenholmePrime.wolstenholme_bernoulli":{"url":"/FormalConjectures/Wikipedia/WolstenholmePrime/#WolstenholmePrime___wolstenholme_bernoulli","anchor":"WolstenholmePrime___wolstenholme_bernoulli","docHtml":"\n Equivalently, a prime $p > 7$ is a Wolstenholme prime if it divides the numerator of the Bernoulli number $B_{p-3}$.
"},"WolstenholmePrime.wolstenholme_harmonic":{"url":"/FormalConjectures/Wikipedia/WolstenholmePrime/#WolstenholmePrime___wolstenholme_harmonic","anchor":"WolstenholmePrime___wolstenholme_harmonic","docHtml":"\n Another equivalent definition is that a prime $p > 7$ is a Wolstenholme prime\nif it $p^3$ divides the numerator of the harmonic number $H_{p-1}$.
"},"WolstenholmePrime.wolstenholme_prime_infinite":{"url":"/FormalConjectures/Wikipedia/WolstenholmePrime/#WolstenholmePrime___wolstenholme_prime_infinite","anchor":"WolstenholmePrime___wolstenholme_prime_infinite","docHtml":"\n It is conjectured that there are infinitely many Wolstenholme primes.
\n\n
\n Consider the following operation on the natural numbers:\nIf the number is even, take the floor of the square root.\nIf the number is odd, take the floor of n raised to the 3/2 power.
"},"JugglerConjecture.juggler_conjecture":{"url":"/FormalConjectures/Wikipedia/JugglerConjecture/#JugglerConjecture___juggler_conjecture","anchor":"JugglerConjecture___juggler_conjecture","docHtml":"\n Now form a sequence beginning with any positive integer, where each subsequent term is obtained\nby applying the operation defined above to the previous term.\nThe Juggler Conjecture states that for any positive integer $n$, there exists a natural number\n$m$ such that the $m$-th term of the sequence is $1$.
"},"JugglerConjecture.jugglerStep_36":{"url":"/FormalConjectures/Wikipedia/JugglerConjecture/#JugglerConjecture___jugglerStep_36","anchor":"JugglerConjecture___jugglerStep_36","docHtml":"\n Example: jugglerStep 36 = ⌊36^(1/2)⌋ = ⌊6⌋ = 6 (since 36 is even).
"},"InscribedSquare.IsRectangle":{"url":"/FormalConjectures/Wikipedia/InscribedSquare/#InscribedSquare___IsRectangle","anchor":"InscribedSquare___IsRectangle","docHtml":"\n Four points a b c d in the plane form a rectangle with a opposite to c iff the line\nsegments from a to c and from b to d have both the same length and the same midpoint, acting\nas the diagonals of the rectangle. We also require the rectangle to be nondegenerate and have a\ngiven aspect ratio ratio : ℝ.
\n Four points a b c d in the plane form a rectangle with a opposite to c iff the line\nsegments from a to c and from b to d have both the same length and the same midpoint, acting\nas the diagonals of the rectangle. We also require the rectangle to be nondegenerate and have a\ngiven aspect ratio ratio : ℝ.
\n Four points a b c d in the plane form a rectangle with a opposite to c iff the line\nsegments from a to c and from b to d have both the same length and the same midpoint, acting\nas the diagonals of the rectangle. We also require the rectangle to be nondegenerate and have a\ngiven aspect ratio ratio : ℝ.
\n Four points a b c d in the plane form a rectangle with a opposite to c iff the line\nsegments from a to c and from b to d have both the same length and the same midpoint, acting\nas the diagonals of the rectangle. We also require the rectangle to be nondegenerate and have a\ngiven aspect ratio ratio : ℝ.
\n Four points a b c d in the plane form a rectangle with a opposite to c iff the line\nsegments from a to c and from b to d have both the same length and the same midpoint, acting\nas the diagonals of the rectangle. We also require the rectangle to be nondegenerate and have a\ngiven aspect ratio ratio : ℝ.
\n Four points a b c d in the plane form a rectangle with a opposite to c iff the line\nsegments from a to c and from b to d have both the same length and the same midpoint, acting\nas the diagonals of the rectangle. We also require the rectangle to be nondegenerate and have a\ngiven aspect ratio ratio : ℝ.
\nInscribed square problem\nDoes every Jordan curve admit an inscribed square?
"},"InscribedSquare.inscribed_rectangle_problem":{"url":"/FormalConjectures/Wikipedia/InscribedSquare/#InscribedSquare___inscribed_rectangle_problem","anchor":"InscribedSquare___inscribed_rectangle_problem","docHtml":"\nInscribed rectangle problem\nDoes every Jordan curve admit inscribed rectangles of any given aspect ratio?
"},"InscribedSquare.exists_inscribed_rectangle":{"url":"/FormalConjectures/Wikipedia/InscribedSquare/#InscribedSquare___exists_inscribed_rectangle","anchor":"InscribedSquare___exists_inscribed_rectangle","docHtml":"\n It is known that every Jordan curve admits at least one inscribed rectangle.
"},"InscribedSquare.exists_inscribed_rectangle_of_smooth":{"url":"/FormalConjectures/Wikipedia/InscribedSquare/#InscribedSquare___exists_inscribed_rectangle_of_smooth","anchor":"InscribedSquare___exists_inscribed_rectangle_of_smooth","docHtml":"\n It is known that every
\n It is also known that every $C^2$ Jordan curve admits an inscribed square.
"},"BetrothedNumbers.IsBetrothed":{"url":"/FormalConjectures/Wikipedia/BetrothedNumbers/#BetrothedNumbers___IsBetrothed","anchor":"BetrothedNumbers___IsBetrothed","docHtml":"\n Two natural numbers $m$ and $n$ are betrothed (or quasi-amicable) if $\\sigma(m) = \\sigma(n) = m + n + 1$,\nwhere $\\sigma$ is the sum-of-divisors function. Equivalently, the sum of the proper divisors\nof $m$ equals $n + 1$, and the sum of the proper divisors of $n$ equals $m + 1$.
"},"BetrothedNumbers.IsBetrothed.left":{"url":"/FormalConjectures/Wikipedia/BetrothedNumbers/#BetrothedNumbers___IsBetrothed___left","anchor":"BetrothedNumbers___IsBetrothed___left","docHtml":"\n Two natural numbers $m$ and $n$ are betrothed (or quasi-amicable) if $\\sigma(m) = \\sigma(n) = m + n + 1$,\nwhere $\\sigma$ is the sum-of-divisors function. Equivalently, the sum of the proper divisors\nof $m$ equals $n + 1$, and the sum of the proper divisors of $n$ equals $m + 1$.
"},"BetrothedNumbers.IsBetrothed.right":{"url":"/FormalConjectures/Wikipedia/BetrothedNumbers/#BetrothedNumbers___IsBetrothed___right","anchor":"BetrothedNumbers___IsBetrothed___right","docHtml":"\n Two natural numbers $m$ and $n$ are betrothed (or quasi-amicable) if $\\sigma(m) = \\sigma(n) = m + n + 1$,\nwhere $\\sigma$ is the sum-of-divisors function. Equivalently, the sum of the proper divisors\nof $m$ equals $n + 1$, and the sum of the proper divisors of $n$ equals $m + 1$.
"},"BetrothedNumbers.betrothed_48_75":{"url":"/FormalConjectures/Wikipedia/BetrothedNumbers/#BetrothedNumbers___betrothed_48_75","anchor":"BetrothedNumbers___betrothed_48_75","docHtml":"\n The smallest known betrothed pair $(48, 75)$.
"},"BetrothedNumbers.IsBetrothed.symm":{"url":"/FormalConjectures/Wikipedia/BetrothedNumbers/#BetrothedNumbers___IsBetrothed___symm","anchor":"BetrothedNumbers___IsBetrothed___symm","docHtml":"\nIsBetrothed is symmetric.
\nSame parity betrothed numbers conjecture.\nDo there exist betrothed numbers $(m, n)$ where both have the same parity\n(both even or both odd)?
\n\n All known betrothed pairs consist of one even and one odd number.
"},"BetrothedNumbers.infinitely_many_betrothed":{"url":"/FormalConjectures/Wikipedia/BetrothedNumbers/#BetrothedNumbers___infinitely_many_betrothed","anchor":"BetrothedNumbers___infinitely_many_betrothed","docHtml":"\nInfinitude of betrothed numbers conjecture.\nAre there infinitely many betrothed number pairs?
"},"BoundedBurnsideProblem.bounded_burnside_problem":{"url":"/FormalConjectures/Wikipedia/BoundedBurnsideProblem/#BoundedBurnsideProblem___bounded_burnside_problem","anchor":"BoundedBurnsideProblem___bounded_burnside_problem","docHtml":"\n Let $G$ be a finitely generated group, and assume there exists $n$ such that for every $g$ in $G$,\n$g^n = 1$. Is $G$ necessarily finite?
"},"GottschalkSurjunctivity.shift":{"url":"/FormalConjectures/Wikipedia/SurjunctiveGroup/#GottschalkSurjunctivity___shift","anchor":"GottschalkSurjunctivity___shift","docHtml":"\n The left shift of x : G → A by g : G, defined by (shift g x)(h) = x(g⁻¹ * h).\nThis is the standard left shift action of G on A^G. We define it as a plain function\nrather than a MulAction instance to avoid conflict with the pointwise Pi.instMulAction.
\n A map τ : (G → A) → (G → A) is τ(shift g x) = shift g (τ x) for all g : G and x : G → A.
\n A group G is A, every injective,\ncontinuous, shift-equivariant map (G → A) → (G → A) is also surjective.
\n Continuity is with respect to the product topology on G → A where A carries the\ndiscrete topology.
\nGottschalk's surjunctivity conjecture (1973): every group is surjunctive.\nThat is, for every group G and every finite alphabet A, every injective cellular\nautomaton on A^G is surjective.
\n Every finite group is surjunctive. This is a classical result: an injective\nendomorphism of a finite set is surjective.
"},"Mahler32.Ω":{"url":"/FormalConjectures/Wikipedia/Mahler32/#Mahler32______","anchor":"Mahler32______","docHtml":"\n For a real number α, define Ω(α) as\n$$\n\\Omega (\\alpha )=\\inf _{\\theta > 0}\\left({\\limsup _{n\\rightarrow \\infty }\\left\\lbrace\n{\\theta \\alpha ^{n}}\\right\\rbrace -\\liminf _{n\\rightarrow \\infty }\\left\\lbrace {\\theta \\alpha ^{n}}\\right\\rbrace }\\right).\n$$
\n A Z-number is a real number x such that the fractional parts of x(3/2)^n are less than\n1/2 for all positive integers n.
\n The Mahler Conjecture states that there are no non-zero Z-numbers.
"},"Mahler32.mahler_conjecture.variants.consequence":{"url":"/FormalConjectures/Wikipedia/Mahler32/#Mahler32___mahler_conjecture___variants___consequence","anchor":"Mahler32___mahler_conjecture___variants___consequence","docHtml":"\n If Mahler's conjecture is true, i.e. there are no Z-numbers, then Ω(3/2) exceeds 1/2.
\n It is known that for all rational p/q > 1 in lowest terms, we have Ω(p/q) > 1/p.
\n The radical of n denoted is the product of the distinct prime factors of n.
\n Quality q(a, b, c) of the triple (a, b, c) is defined as q(a,b,c) = log (c) / log (rad(abc)).
\n For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that c > rad(abc)^(1+ε)
\n For every positive real number ε, there exists a constant K_ε such that for all triples (a, b, c) of coprime positive integers, with a + b = c we have c < K_ε rad(abc)^(1+ε).
\n For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε.
\n $x$ is a candidate for being a taxicab number for $k, m, n$\nif there exists a (finite) set of at least $n$ distinct,\npairwise disjoint, non-empty, non-zero lists of length\n$m$, such that the sum of the $k$-th powers of the\nelements of each list is $x$. The disjointness condition\nensures that the representations do not share any common terms.
"},"Taxicab.taxicab_1729":{"url":"/FormalConjectures/Wikipedia/Taxicab/#Taxicab___taxicab_1729","anchor":"Taxicab___taxicab_1729","docHtml":"\n $1729$ is a possible taxicab number for $k=3, m=2, n=2$.
"},"Taxicab.IsTaxicabFor":{"url":"/FormalConjectures/Wikipedia/Taxicab/#Taxicab___IsTaxicabFor","anchor":"Taxicab___IsTaxicabFor","docHtml":"\n $x$ is a taxicab number if it is the smallest number\nthat can be expressed as a sum of $m$ positive $k$-th\npowers in at least $n$ distinct ways.
"},"Taxicab.taxicab_4'":{"url":"/FormalConjectures/Wikipedia/Taxicab/#Taxicab___taxicab_4___","anchor":"Taxicab___taxicab_4___"},"Taxicab.taxicab_4":{"url":"/FormalConjectures/Wikipedia/Taxicab/#Taxicab___taxicab_4","anchor":"Taxicab___taxicab_4","docHtml":"\n Using Aristotle (Harmonic) we get a compact proof that\n4 is the taxicab number for $k=1, m=2, n=2$.
"},"Taxicab.taxicab_for_5_2_2":{"url":"/FormalConjectures/Wikipedia/Taxicab/#Taxicab___taxicab_for_5_2_2","anchor":"Taxicab___taxicab_for_5_2_2","docHtml":"\n Taxicab number for $k=5$, $m=2$, and $n=2$ is not known.\nWhether such a number exists is also not known.
"},"Taxicab.taxicab_for_5_2_n":{"url":"/FormalConjectures/Wikipedia/Taxicab/#Taxicab___taxicab_for_5_2_n","anchor":"Taxicab___taxicab_for_5_2_n","docHtml":"\n Taxicab number for $k=5$ and $m=2$ is not-known for any $n ≥ 2$.\nWhether such a number exists is also not known.
"},"SnakeInBox.Hypercube":{"url":"/FormalConjectures/Wikipedia/SnakeInTheBox/#SnakeInBox___Hypercube","anchor":"SnakeInBox___Hypercube","docHtml":"\n A graph on the power set of Fin n, where two sets are adjacent if they differ by a single element.
\n A subgraph G' is a 'snake' of length k in graph G if it is an induced path of length k.
\n The length of the longest induced path (or 'snake') in a graph G.
\n The length of the longest snake for the Hypercube n graph.
\n The longest snake in the $0$-dimensional cube, i.e. the cube consisting of one point, is zero,\nsince there only is one induced path and it is of length zero.
"},"SnakeInBox.snake_small_dimensions":{"url":"/FormalConjectures/Wikipedia/SnakeInTheBox/#SnakeInBox___snake_small_dimensions","anchor":"SnakeInBox___snake_small_dimensions","docHtml":"\n The maximum length for the snake-in-the-box problem is known for dimensions zero through eight;\nit is $0, 1, 2, 4, 7, 13, 26, 50, 98$.
"},"SnakeInBox.snake_dim_nine":{"url":"/FormalConjectures/Wikipedia/SnakeInTheBox/#SnakeInBox___snake_dim_nine","anchor":"SnakeInBox___snake_dim_nine","docHtml":"\n For dimension $9$, the length of the longest snake in the box is not known.\nThis is currently the smallest dimension where this question is open.
"},"SnakeInBox.snake_dim_nine_lower_bound":{"url":"/FormalConjectures/Wikipedia/SnakeInTheBox/#SnakeInBox___snake_dim_nine_lower_bound","anchor":"SnakeInBox___snake_dim_nine_lower_bound","docHtml":"\n The best length found so far for dimension nine is 190.
"},"SnakeInBox.snake_upper_bound":{"url":"/FormalConjectures/Wikipedia/SnakeInTheBox/#SnakeInBox___snake_upper_bound","anchor":"SnakeInBox___snake_upper_bound","docHtml":"\n An upper bound of the maximal length of the longest snake in a box is given by\n$$\n1 + 2^{n-1}\\frac{6n}{6n + \\frac{1}{6\\sqrt{6}}n^{\\frac 1 2} - 7}.\n$$
"},"Mersenne.Nat.GivesWagstaffPrime":{"url":"/FormalConjectures/Wikipedia/Mersenne/#Mersenne___Nat___GivesWagstaffPrime","anchor":"Mersenne___Nat___GivesWagstaffPrime","docHtml":"\n A Wagstaff prime is a prime number of the form $(2^p+1)/3$.
"},"Mersenne.Nat.IsSpecialForm":{"url":"/FormalConjectures/Wikipedia/Mersenne/#Mersenne___Nat___IsSpecialForm","anchor":"Mersenne___Nat___IsSpecialForm","docHtml":"\n Holds when there is exists a number k such that $p = 2^k \\pm 1$ or $p = 4^k \\pm 3$.
\n The Catalan-Mersenne numbers, defined recursively by $c_0 = 2$ and\n$c_{n+1} = 2^{c_n} - 1$.
"},"Mersenne.NewMersenneConjectureStatement":{"url":"/FormalConjectures/Wikipedia/Mersenne/#Mersenne___NewMersenneConjectureStatement","anchor":"Mersenne___NewMersenneConjectureStatement","docHtml":"\n A natural number p satisfies the statement of the New Mersenne Conjecture if whenever\ntwo of the following conditions hold,\nthen all three must hold:
\n $2^p-1$ is prime
\n\n $(2^p+1)/3$ is prime
\n\n Exists a number k such that $p = 2^k \\pm 1$ or $p = 4^k \\pm 3$
\n For any odd natural number p if two of the following conditions hold,\nthen all three must hold:
\n $2^p-1$ is prime
\n\n $(2^p+1)/3$ is prime
\n\n Exists a number k such that $p = 2^k \\pm 1$ or $p = 4^k \\pm 3$
\n It suffices to check this conjecture for primes
"},"Mersenne.new_mersenne_conjecture.variants.prime":{"url":"/FormalConjectures/Wikipedia/Mersenne/#Mersenne___new_mersenne_conjecture___variants___prime","anchor":"Mersenne___new_mersenne_conjecture___variants___prime","docHtml":"\n The New Mersenne Conjecture statement holds for odd primes.
"},"Mersenne.infinitely_many_mersenne_primes":{"url":"/FormalConjectures/Wikipedia/Mersenne/#Mersenne___infinitely_many_mersenne_primes","anchor":"Mersenne___infinitely_many_mersenne_primes","docHtml":"\n Are there infinitely many Mersenne primes?
"},"Mersenne.catalans_mersenne_conjecture":{"url":"/FormalConjectures/Wikipedia/Mersenne/#Mersenne___catalans_mersenne_conjecture","anchor":"Mersenne___catalans_mersenne_conjecture","docHtml":"\n The first five Catalan-Mersenne numbers $c_0, \\ldots, c_4$ are known to be prime.\nCatalan conjectured that they are prime \"up to a certain limit\".\nAre all Catalan-Mersenne numbers $c_n$ with $n \\geq 5$ prime?
"},"SumOfThreeCubes.IsSumOfThreeCubes":{"url":"/FormalConjectures/Wikipedia/SumOfThreeCubes/#SumOfThreeCubes___IsSumOfThreeCubes","anchor":"SumOfThreeCubes___IsSumOfThreeCubes","docHtml":"\n The predicate that n : R is a sum of three cubes.
\n Any rational number is a sum of three rational cubes.
\n\n First proved by Ryley in 1825, which can be found in [Ri1930].\nThe below parametrization is brought from the MSE answer [MSE].
\n\n [Ri1930] Richmond, H. W. \"On Rational Solutions of $x^3 + y^3 + z^3 = R$.\" Proceedings of the Edinburgh Mathematical Society 2.2 (1930): 92-100.\n[MSE] Kieren MacMillan, Proving that any rational number can be represented as the sum of the cubes of three rational numbers, https://math.stackexchange.com/q/4480969
"},"SumOfThreeCubes.isSumOfThreeCubes_iff_mod_9":{"url":"/FormalConjectures/Wikipedia/SumOfThreeCubes/#SumOfThreeCubes___isSumOfThreeCubes_iff_mod_9","anchor":"SumOfThreeCubes___isSumOfThreeCubes_iff_mod_9","docHtml":"\n An integer n : ℤ can be written as a sum of three cubes (of integers) if and only if\nn is not 4 or 5 mod 9.
\n Are $e$ and $\\pi$ algebraically independent?
"},"Irrational.irrational_e_plus_pi":{"url":"/FormalConjectures/Wikipedia/Irrational/#Irrational___irrational_e_plus_pi","anchor":"Irrational___irrational_e_plus_pi","docHtml":"\n Is $e + \\pi$ irrational?
"},"Irrational.irrational_e_times_pi":{"url":"/FormalConjectures/Wikipedia/Irrational/#Irrational___irrational_e_times_pi","anchor":"Irrational___irrational_e_times_pi","docHtml":"\n Is $e \\pi$ irrational?
"},"Irrational.irrational_e_to_e":{"url":"/FormalConjectures/Wikipedia/Irrational/#Irrational___irrational_e_to_e","anchor":"Irrational___irrational_e_to_e","docHtml":"\n Is $e ^ e$ irrational?
"},"Irrational.irrational_pi_to_e":{"url":"/FormalConjectures/Wikipedia/Irrational/#Irrational___irrational_pi_to_e","anchor":"Irrational___irrational_pi_to_e","docHtml":"\n Is $\\pi ^ e$ irrational?
"},"Irrational.irrational_pi_to_pi":{"url":"/FormalConjectures/Wikipedia/Irrational/#Irrational___irrational_pi_to_pi","anchor":"Irrational___irrational_pi_to_pi","docHtml":"\n Is $\\pi ^ \\pi$ irrational?
"},"Irrational.irrational_ln_pi":{"url":"/FormalConjectures/Wikipedia/Irrational/#Irrational___irrational_ln_pi","anchor":"Irrational___irrational_ln_pi","docHtml":"\n Is $\\ln(\\pi)$ irrational?
"},"Irrational.irrational_eulerMascheroniConstant":{"url":"/FormalConjectures/Wikipedia/Irrational/#Irrational___irrational_eulerMascheroniConstant","anchor":"Irrational___irrational_eulerMascheroniConstant","docHtml":"\n Is the Euler-Mascheroni constant $\\gamma$ irrational?
"},"Irrational.irrational_catalanConstant":{"url":"/FormalConjectures/Wikipedia/Irrational/#Irrational___irrational_catalanConstant","anchor":"Irrational___irrational_catalanConstant","docHtml":"\n Is the Catalan constant $$G = \\sum_{n=0}^∞ (-1)^n / (2n + 1)^2 \\approx 0.91596$$ irrational?
"},"FibonacciPrimes.fib_primes_infinite":{"url":"/FormalConjectures/Wikipedia/FibonacciPrimes/#FibonacciPrimes___fib_primes_infinite","anchor":"FibonacciPrimes___fib_primes_infinite","docHtml":"\n There are infinitely many Fibonacci primes, i.e., Fibonacci numbers that are prime\nIt is also a barrier to defining a benchmark from this paper:\nhttps://arxiv.org/html/2505.13938v1 (see Figure 8).
"},"FibonacciPrimes.fib_primes_infinite.variant":{"url":"/FormalConjectures/Wikipedia/FibonacciPrimes/#FibonacciPrimes___fib_primes_infinite___variant","anchor":"FibonacciPrimes___fib_primes_infinite___variant","docHtml":"\n There are infinitely many indices $i$, such that the $i$-th Fibonacci is prime.
"},"FibonacciPrimes.indices_infinite_iff_fib_primes_infinite":{"url":"/FormalConjectures/Wikipedia/FibonacciPrimes/#FibonacciPrimes___indices_infinite_iff_fib_primes_infinite","anchor":"FibonacciPrimes___indices_infinite_iff_fib_primes_infinite","docHtml":"\n The two ways of phrasing the conjecture are equivalent.
"},"TwinPrimes.twin_primes":{"url":"/FormalConjectures/Wikipedia/TwinPrimes/#TwinPrimes___twin_primes","anchor":"TwinPrimes___twin_primes","docHtml":"\n Are there infinitely many primes p such that p + 2 is prime?
"},"PerfectNumbers.infinitely_many_perfect":{"url":"/FormalConjectures/Wikipedia/PerfectNumbers/#PerfectNumbers___infinitely_many_perfect","anchor":"PerfectNumbers___infinitely_many_perfect","docHtml":"\nInfinitely many perfect numbers conjecture.\nAre there infinitely many perfect numbers?
\n\n
\nInfinitely many even perfect numbers conjecture.\nAre there infinitely many even perfect numbers?
\n\n This is equivalent to asking whether there are infinitely many Mersenne primes,\nsince by the Euclid–Euler theorem an even number is perfect if and only if it\nhas the form $2^{p-1}(2^p - 1)$ where $2^p - 1$ is a Mersenne prime.
\n\n
\nOdd Perfect Number Conjecture.\nThe Odd Perfect Number Conjecture states that all perfect numbers are even.
\n\n
\n A known result: If an odd perfect number exists, it must be greater than $10^{1500}$\nand must have at least 101 prime factors (including multiplicities).
\n\n
\n A known result: If an odd perfect number exists, it must be of the form\n$p^α * m^2$ where $p$ is prime, $p \\equiv 1 \\pmod{4}$, $\\alpha \\equiv 1 \\pmod{4}$,\nand $p \\nmid m$.
\n\n
\n Formal proof linked here provided by AlphaProof.
"},"Grimm.grimm_conjecture":{"url":"/FormalConjectures/Wikipedia/Grimm/#Grimm___grimm_conjecture","anchor":"Grimm___grimm_conjecture","docHtml":"\nGrimm's Conjecture\nIf $n, n+1, \\dots, n+k-1$ are all composite numbers, then there are $k$ distinct primes $p_i$\nsuch that $p_i$ divides $n + i$ for all $0 \\le i \\le k-1$.
"},"Grimm.grimm_conjecture_weak":{"url":"/FormalConjectures/Wikipedia/Grimm/#Grimm___grimm_conjecture_weak","anchor":"Grimm___grimm_conjecture_weak","docHtml":"\nGrimm's Conjecture, weaker version\nIf $n, n+1, \\dots, n+k-1$ are all composite numbers, then their product\nhas at least $k$ distinct prime divisors.
"},"QuasiperfectNumbers.Quasiperfect":{"url":"/FormalConjectures/Wikipedia/QuasiperfectNumbers/#QuasiperfectNumbers___Quasiperfect","anchor":"QuasiperfectNumbers___Quasiperfect","docHtml":"\n A number is quasiperfect if the sum of its divisors is equal to $2n + 1$.
"},"QuasiperfectNumbers.exists_quasiperfect":{"url":"/FormalConjectures/Wikipedia/QuasiperfectNumbers/#QuasiperfectNumbers___exists_quasiperfect","anchor":"QuasiperfectNumbers___exists_quasiperfect","docHtml":"\nQuasiperfect Numbers Conjecture.\nDo quasiperfect numbers exist?
"},"LanderParkinSelfridge.lander_parkin_selfridge":{"url":"/FormalConjectures/Wikipedia/LanderParkinAndSelfridgeConjecture/#LanderParkinSelfridge___lander_parkin_selfridge","anchor":"LanderParkinSelfridge___lander_parkin_selfridge","docHtml":"\n The Lander–Parkin–Selfridge conjecture: if the sum of $n$ positive integer $k$-th powers\nequals the sum of $m$ positive integer $k$-th powers, with all values on the left distinct from\nall values on the right, then $n + m \\geq k$.
\n\n Formally, for positive integers $k, n, m \\in \\mathbb{N}$ and sequences\n$x : {0, \\ldots, n-1} \\to \\mathbb{N}$ and $y : {0, \\ldots, m-1} \\to \\mathbb{N}$\nwith $x_i > 0$, $y_j > 0$, and $x_i \\neq y_j$ for all $i, j$, if\n$$\\sum_{i=0}^{n-1} x_i^k = \\sum_{j=0}^{m-1} y_j^k,$$\nthen $k \\leq n + m$.
"},"LanderParkinSelfridge.lander_parkin_selfridge.variants.five_three":{"url":"/FormalConjectures/Wikipedia/LanderParkinAndSelfridgeConjecture/#LanderParkinSelfridge___lander_parkin_selfridge___variants___five_three","anchor":"LanderParkinSelfridge___lander_parkin_selfridge___variants___five_three","docHtml":"\n Special case of the Lander–Parkin–Selfridge conjecture: there is no solution in positive\nintegers to\n$$x_1^5 + x_2^5 + x_3^5 = y^5.$$\nThat is, for all $x_1, x_2, x_3, y \\in \\mathbb{N}$ with $x_1, x_2, x_3, y > 0$,\n$$x_1^5 + x_2^5 + x_3^5 \\neq y^5.$$\nThis corresponds to the case $k = 5$, $n = 3$, $m = 1$ of the general conjecture,\nwhere $n + m = 4 < 5 = k$ would be required to yield a counterexample.
"},"Sendov.Polynomial.IsSendov":{"url":"/FormalConjectures/Wikipedia/Sendov/#Sendov___Polynomial___IsSendov","anchor":"Sendov___Polynomial___IsSendov","docHtml":"\n The predicate that a polynomial satisfies the hypotheses of Sendov's conjecture.
\n\nf.IsSendov holds if f has degree at least 2 and all roots of f lie in the unit disc\nof the complex plane.
\nSatisfiesSendovConjecture n states that Sendov's conjecture is true for every polynomial of\ndegree n.
\nSendov's conjecture states that for a polynomial\n$$f(z)=(z-r_{1})\\cdots (z-r_{n}),\\qquad (n\\geq 2)$$\nwith all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a\ndistance no more than $1$ from at least one critical point.
"},"Sendov.sendov_conjecture.variants.le_nine":{"url":"/FormalConjectures/Wikipedia/Sendov/#Sendov___sendov_conjecture___variants___le_nine","anchor":"Sendov___sendov_conjecture___variants___le_nine","docHtml":"\nSendov's conjecture states that for a polynomial\n$$f(z)=(z-r_{1})\\cdots (z-r_{n}),\\qquad (n\\geq 2)$$\nwith all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a\ndistance no more than $1$ from at least one critical point.
\n\n It has been shown that Sendov's conjecture holds when the degree of $n$ is at most $9$.
"},"Sendov.sendov_conjecture.variants.eventually_true":{"url":"/FormalConjectures/Wikipedia/Sendov/#Sendov___sendov_conjecture___variants___eventually_true","anchor":"Sendov___sendov_conjecture___variants___eventually_true","docHtml":"\nSendov's conjecture states that for a polynomial\n$$f(z)=(z-r_{1})\\cdots (z-r_{n}),\\qquad (n\\geq 2)$$\nwith all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a\ndistance no more than $1$ from at least one critical point.
\n\n It has been shown that Sendov's conjecture holds for polynomials of sufficiently large degree.
"},"Lemoine.OddPrime":{"url":"/FormalConjectures/Wikipedia/Lemoine/#Lemoine___OddPrime","anchor":"Lemoine___OddPrime","docHtml":"\n
\n [Ki85] Kiltinen, J. and Young P. (1985). Goldbach, Lemoine, and a Know/Don't Know Problem.
\n\n For all odd integers $n ≥ 7$ there are prime numbers $p,q$ such that $n = p+2q$.
"},"Lemoine.lemoine_conjecture_extension":{"url":"/FormalConjectures/Wikipedia/Lemoine/#Lemoine___lemoine_conjecture_extension","anchor":"Lemoine___lemoine_conjecture_extension","docHtml":"\n For all odd integers $n ≥ 9$ there are odd prime numbers $p,q,r,s$ and natural numbers $a,b$\nsuch that $p+2q = n$, $2+pq = 2^a+r$, $2p+q = 2^b+s$
"},"MeanValueProblem.mean_value_problem":{"url":"/FormalConjectures/Wikipedia/MeanValueProblem/#MeanValueProblem___mean_value_problem","anchor":"MeanValueProblem___mean_value_problem","docHtml":"\n Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$\nthere is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ |p'(z)|$.
"},"MeanValueProblem.mean_value_problem_leq_4":{"url":"/FormalConjectures/Wikipedia/MeanValueProblem/#MeanValueProblem___mean_value_problem_leq_4","anchor":"MeanValueProblem___mean_value_problem_leq_4","docHtml":"\n The following weaker version of the mean value problem has been proven.\nGiven a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$\nthere a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ 4|p'(z)|$.
"},"MeanValueProblem.mean_value_problem_of_real_roots":{"url":"/FormalConjectures/Wikipedia/MeanValueProblem/#MeanValueProblem___mean_value_problem_of_real_roots","anchor":"MeanValueProblem___mean_value_problem_of_real_roots","docHtml":"\n The following tighter bound depending on the degree $d$ of the polynomial $p$,\nin the case of $p$ only having real roots has been shown by Tischler.\n$|p(z)-p(c)|/|z-c| \\le (d-1)/d \\cdot |p'(z)|$
"},"MeanValueProblem.mean_value_problem_of_roots_same_norm":{"url":"/FormalConjectures/Wikipedia/MeanValueProblem/#MeanValueProblem___mean_value_problem_of_roots_same_norm","anchor":"MeanValueProblem___mean_value_problem_of_roots_same_norm","docHtml":"\n The following tighter bound depending on the degree $d$ of the polynomial $p$,\nin the case of $p$ all roots having the same norm has been shown by Tischler.\n$|p(z) - p(c)|/|z-c| \\le (d-1)/d \\cdot |p'(z)|$.
"},"GaussCircleProblem.N":{"url":"/FormalConjectures/Wikipedia/GaussCircleProblem/#GaussCircleProblem___N","anchor":"GaussCircleProblem___N","docHtml":"\n Let $N(r)$ be the number of points $(m, n)$ within a circle of radius $r$,\nwhere $m$ and $n$ are both integers.
"},"GaussCircleProblem.E":{"url":"/FormalConjectures/Wikipedia/GaussCircleProblem/#GaussCircleProblem___E","anchor":"GaussCircleProblem___E","docHtml":"\n Let $E(r)$ be the error term between the number of integral points inside the circle and the\narea of the circle; that is $N(r) = \\pi r^2 + E(r)$.
"},"GaussCircleProblem.error_le":{"url":"/FormalConjectures/Wikipedia/GaussCircleProblem/#GaussCircleProblem___error_le","anchor":"GaussCircleProblem___error_le","docHtml":"\n Gauss proved that\n$$\n|E(r)|\\leq 2\\sqrt{2}\\pi r,\n$$\nfor sufficiently large $r$.
\n\n [Ha59] Hardy, G. H. (1959).
\n Hardy and Laundau independently found a lower bound by showing that\n$$\n|E(r)| \\neq o\\left(r^{1/2}(\\log r)^{1/4}\\right)\n$$
"},"GaussCircleProblem.error_isBigO":{"url":"/FormalConjectures/Wikipedia/GaussCircleProblem/#GaussCircleProblem___error_isBigO","anchor":"GaussCircleProblem___error_isBigO","docHtml":"\n It is conjectured that the correct bound is\n$$\n|E(r)| = O\\left(r^{1/2 + o(1)}\\right)\n$$
\n\n [Ha59] Hardy, G. H. (1959).
\n See also https://arxiv.org/abs/2305.03549
"},"GaussCircleProblem.exact_form_floor":{"url":"/FormalConjectures/Wikipedia/GaussCircleProblem/#GaussCircleProblem___exact_form_floor","anchor":"GaussCircleProblem___exact_form_floor","docHtml":"\n The value of $N(r)$ can be expressed as\n$$\nN(r) = 1 + 4\\sum_{i=0}^{\\infty}\\left(\\left\\lfloor\\frac{r^2}{4i+1}\\right\\rfloor -\n\\left\\lfloor\\frac{r^2}{4i + 3}\\right\\rfloor\\right).\n$$
"},"LonelyRunnerConjecture.lonely_runner_conjecture":{"url":"/FormalConjectures/Wikipedia/LonelyRunnerConjecture/#LonelyRunnerConjecture___lonely_runner_conjecture","anchor":"LonelyRunnerConjecture___lonely_runner_conjecture","docHtml":"\n Consider $n$ runners on a circular track of unit length. At the initial time\n$t = 0$, all runners are at the same position and start to run; the runners'\nspeeds are constant, all distinct, and may be negative. A runner is said to be\nlonely at time $t$ if they are at a distance (measured along the circle) of at\nleast $\\frac 1 n$ from every other runner. The lonely runner conjecture states that each\nrunner is lonely at some time, no matter the choice of speeds.
"},"LonelyRunnerConjecture.deltaTuple":{"url":"/FormalConjectures/Wikipedia/LonelyRunnerConjecture/#LonelyRunnerConjecture___deltaTuple","anchor":"LonelyRunnerConjecture___deltaTuple","docHtml":"\n For an $n$-tuple of distinct integer velocities $v_1,\\dots,v_n$,\ndeltaTuple v is the maximal value of $\\min_i |t v_i|_{\\mathbb{R}/\\mathbb{Z}}$ over time.
\n The $n$th deltaTuple\nover all $n$-tuples of distinct nonzero integer velocities.
\nTheorem 1.3 (Tao, 2017; arXiv:1701.02048).\nThere exists an absolute constant $c > 0$ such that for all sufficiently large $n$,\nthe gap of loneliness satisfies\n$\\delta_n \\ge \\frac{1}{2n} + \\frac{c \\log n}{n^2 (\\log \\log n)^2}$.
"},"Kakeya.IsKakeya":{"url":"/FormalConjectures/Wikipedia/Kakeya/#Kakeya___IsKakeya","anchor":"Kakeya___IsKakeya","docHtml":"\n A set S in ℝⁿ is called a Kakeya set if it contains a unit line segment in every direction.\nFor simplicity, we omit the compactness assumption here.\nFor a discussion on the equivalence of definitions with and without compactness, see\nthis paper.
\n A trivial example: the closed ball of radius 1 in ℝⁿ is a Kakeya set.
\n The Kakeya set conjecture in dimension n: the statement that every Kakeya set in ℝⁿ has\nHausdorff dimension n.
\n The Kakeya set conjecture: Kakeya sets in $\\mathbb{R}^n$ have Hausdorff dimension $n$.
"},"Kakeya.kakeya_2d":{"url":"/FormalConjectures/Wikipedia/Kakeya/#Kakeya___kakeya_2d","anchor":"Kakeya___kakeya_2d","docHtml":"\n The two-dimensional case, proved by Davies [Da71].
\n\n [Da71] Davies, R. O.,
\n The three-dimensional case, proved by Wang, Zahl [WaZa25].
\n\n [WaZa25] Wang, H. and Zahl, J.,
\n A finite field variant of the Kakeya problem considers subsets of 𝔽_qⁿ that contain a line in\nevery direction.
\n The finite field Kakeya conjecture asserts that any Kakeya set in 𝔽_qⁿ has size at\nleast c_n · qⁿ for some constant c_n depending only on n.\nThis was first proved by Dvir [Dv08]. The best known bound to date, due to Bukh and Chao [BuCh21],\nestablishes that any Kakeya set in 𝔽_qⁿ has size at least qⁿ / (2 - 1/q)^(n - 1).
\n [Dv08] Dvir, Z.,
\nDickson's conjecture\nIf a finite set of linear integer forms $f_i(n) = a_i n+b_i$ satisfies Schinzel condition,\nthere exist infinitely many natural numbers $m$ such that $f_i(m)$ are primes for all $i$.
"},"Dickson.polignac_conjecture":{"url":"/FormalConjectures/Wikipedia/Dickson/#Dickson___polignac_conjecture","anchor":"Dickson___polignac_conjecture","docHtml":"\nPolignac's conjecture\nFor any integer $k$ there are infinitely many primes $p$ such that $p + 2k$ is prime.
"},"Dickson.infinite_safe_primes":{"url":"/FormalConjectures/Wikipedia/Dickson/#Dickson___infinite_safe_primes","anchor":"Dickson___infinite_safe_primes","docHtml":"\nThe infinitude of Sophie Germain primes\nThere are infinitely many primes $p$ such that $2p + 1$ is prime.
"},"Dickson.infinite_cousin_primes":{"url":"/FormalConjectures/Wikipedia/Dickson/#Dickson___infinite_cousin_primes","anchor":"Dickson___infinite_cousin_primes","docHtml":"\nThe infinitude of cousin primes\nThere are infinitely many primes $p$ such that $p + 4$ is prime.
"},"Dickson.infinite_sexy_primes":{"url":"/FormalConjectures/Wikipedia/Dickson/#Dickson___infinite_sexy_primes","anchor":"Dickson___infinite_sexy_primes","docHtml":"\nThe infinitude of sexy primes\nThere are infinitely many primes $p$ such that $p + 6$ is prime.
"},"LegendreConjecture.legendre_conjecture":{"url":"/FormalConjectures/Wikipedia/LegendreConjecture/#LegendreConjecture___legendre_conjecture","anchor":"LegendreConjecture___legendre_conjecture","docHtml":"\n Does there always exist at least one prime between consecutive perfect squares?
"},"LegendreConjecture.bounded_gap_legendre":{"url":"/FormalConjectures/Wikipedia/LegendreConjecture/#LegendreConjecture___bounded_gap_legendre","anchor":"LegendreConjecture___bounded_gap_legendre","docHtml":"\n If there exists a constant c > 0 such that\n(n + 1).nth Nat.Prime - n.nth Nat.Prime < (n.nth Nat.Prime) ^ (1 / 2 - c) for all large n,\nthen Legendre's conjecture is asymptotically true.
\n Formal proof linked here provided by AlphaProof.
"},"LegendreConjecture.legendre_conjecture.ferreira_large_n":{"url":"/FormalConjectures/Wikipedia/LegendreConjecture/#LegendreConjecture___legendre_conjecture___ferreira_large_n","anchor":"LegendreConjecture___legendre_conjecture___ferreira_large_n","docHtml":"\n Ferreira proved that the conjecture is true for sufficiently large n.
"},"Idoneal.IsIdoneal":{"url":"/FormalConjectures/Wikipedia/IdonealCompleteness/#Idoneal___IsIdoneal","anchor":"Idoneal___IsIdoneal","docHtml":"\n Equivalent definition: A positive integer $n$ is idoneal if and only if it cannot be written as\n$ab + bc + ac$ for distinct positive integers $a, b,$ and $c$.
"},"Idoneal.knownIdonealNumbers":{"url":"/FormalConjectures/Wikipedia/IdonealCompleteness/#Idoneal___knownIdonealNumbers","anchor":"Idoneal___knownIdonealNumbers","docHtml":"\n The 65 known idoneal numbers that are conjectured to be the only idoneal numbers.
"},"Idoneal.knownIdonealNumbers_are_idoneal":{"url":"/FormalConjectures/Wikipedia/IdonealCompleteness/#Idoneal___knownIdonealNumbers_are_idoneal","anchor":"Idoneal___knownIdonealNumbers_are_idoneal","docHtml":"\n All 65 known idoneal numbers are indeed idoneal.
"},"Idoneal.idoneal_numbers_completeness":{"url":"/FormalConjectures/Wikipedia/IdonealCompleteness/#Idoneal___idoneal_numbers_completeness","anchor":"Idoneal___idoneal_numbers_completeness","docHtml":"\n Idoneal numbers completeness conjecture.
"},"Bloch.blochRadius":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___blochRadius","anchor":"Bloch___blochRadius","docHtml":"\n The Bloch radius $B_f$ of a function $f$ is the radius of the largest univalent disk in the\nimage of the unit disk under $f$.
"},"Bloch.zero_le_blochRadius":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___zero_le_blochRadius","anchor":"Bloch___zero_le_blochRadius"},"Bloch.bddBelow_blochRadius":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___bddBelow_blochRadius","anchor":"Bloch___bddBelow_blochRadius"},"Bloch.dis_add_radius_le_of_ball_subset_ball":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___dis_add_radius_le_of_ball_subset_ball","anchor":"Bloch___dis_add_radius_le_of_ball_subset_ball"},"Bloch.radius_le_of_ball_subset_ball":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___radius_le_of_ball_subset_ball","anchor":"Bloch___radius_le_of_ball_subset_ball"},"Bloch.blochRadius_id_eq_one":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___blochRadius_id_eq_one","anchor":"Bloch___blochRadius_id_eq_one"},"Bloch.landauRadius":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___landauRadius","anchor":"Bloch___landauRadius","docHtml":"\n The Landau radius $L_f$ of a function $f$ is the radius of the largest disk in the image of\nthe unit disk under $f$.
"},"Bloch.blochConstant":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___blochConstant","anchor":"Bloch___blochConstant","docHtml":"\n The Bloch constant $B$ is the infimum of the Bloch radius over all functions holomorphic\nin the unit disk such that $f'(0) = 1$.
"},"Bloch.blochConstant_lower_bound":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___blochConstant_lower_bound","anchor":"Bloch___blochConstant_lower_bound","docHtml":"\n It is proved in [CP96] that the Bloch constant is bounded below by\n$\\sqrt{3}/4 + 2 \\times 10^{-4}$
"},"Bloch.blochConstant_upper_bound":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___blochConstant_upper_bound","anchor":"Bloch___blochConstant_upper_bound","docHtml":"\n It is proved in [AG37] that the Bloch constant is bounded above by\n$\\frac{1}{\\sqrt{1 + \\sqrt{3}}}\\frac{\\Gamma(1/3) \\Gamma(11/12)}{\\Gamma(1/4)}$$.
"},"Bloch.blochConstant_exact_value":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___blochConstant_exact_value","anchor":"Bloch___blochConstant_exact_value","docHtml":"\n Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the\nBloch constant.
"},"Bloch.univalentBlochConstant":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___univalentBlochConstant","anchor":"Bloch___univalentBlochConstant","docHtml":"\n The Univalent Bloch constant $B_u$ is the infimum of the Bloch radius over all univalent\nfunctions in the unit disk such that $f'(0) = 1$.
"},"Bloch.univalentBlochConstant_lower_bound":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___univalentBlochConstant_lower_bound","anchor":"Bloch___univalentBlochConstant_lower_bound","docHtml":"\n It is proved in [Skin2009] that the Univalent Bloch constant is bounded below by $0.5708858$.
"},"Bloch.univalentBlochConstant_upper_bound":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___univalentBlochConstant_upper_bound","anchor":"Bloch___univalentBlochConstant_upper_bound","docHtml":"\n The Univalent Bloch constant is trivially bounded above by the Bloch radius of the identity\nfunction, which is $1$. This is the best upper bound we know according to [OptimizationConstants].
"},"deriv":{"url":"/FormalConjectures/Wikipedia/Bloch/#deriv","anchor":"deriv","docHtml":"\n The Univalent Bloch constant is trivially bounded above by the Bloch radius of the identity\nfunction, which is $1$. This is the best upper bound we know according to [OptimizationConstants].
"},"Bloch.landauConstant":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___landauConstant","anchor":"Bloch___landauConstant","docHtml":"\n The Landau constant $L$ is the infimum of the Landau radius over all functions holomorphic\nin the unit disk such that $f'(0) = 1$.
"},"Bloch.landauConstant_lower_bound":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___landauConstant_lower_bound","anchor":"Bloch___landauConstant_lower_bound","docHtml":"\n It is proved in [Ya95] that the Landau constant is bounded below by $0.5 + 10 ^ {-335}$.
"},"Bloch.landauConstant_upper_bound":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___landauConstant_upper_bound","anchor":"Bloch___landauConstant_upper_bound","docHtml":"\n It is proved in [Ra43] that the Landau constant is bounded above by\n$\\frac{1}{\\sqrt{1 + \\sqrt{3}}}\\frac{\\Gamma(1/3) \\Gamma(5/6)}{\\Gamma(1/6)}$.
"},"Bloch.landauConstant_exact_value":{"url":"/FormalConjectures/Wikipedia/Bloch/#Bloch___landauConstant_exact_value","anchor":"Bloch___landauConstant_exact_value","docHtml":"\n In [Ra43], Rademacher says that he strongly believed that this upper bound is the precise value\nof the Landau constant.
"},"HardyLittlewood.IsPrimeConstellation":{"url":"/FormalConjectures/Wikipedia/HardyLittlewood/#HardyLittlewood___IsPrimeConstellation","anchor":"HardyLittlewood___IsPrimeConstellation","docHtml":"\n A prime constellation is a tuple $(p, p + m_1, \\dots, p + m_k)$ such that the $m_i$ are\nall positive even integers and every entry is a prime number.
"},"HardyLittlewood.IsAdmissiblePrimeConstellation":{"url":"/FormalConjectures/Wikipedia/HardyLittlewood/#HardyLittlewood___IsAdmissiblePrimeConstellation","anchor":"HardyLittlewood___IsAdmissiblePrimeConstellation","docHtml":"\n A prime constellation is said to be admissible if its elements do not form a complete\nset of residue classes with respect to any prime.
"},"HardyLittlewood.Nat.numResidues":{"url":"/FormalConjectures/Wikipedia/HardyLittlewood/#HardyLittlewood___Nat___numResidues","anchor":"HardyLittlewood___Nat___numResidues","docHtml":"\n The number of distinct residue classes amongst a tuple $(m_0, \\dots, m_k)$ for a prime $q$.
"},"HardyLittlewood.Nat.primeTupleCounting":{"url":"/FormalConjectures/Wikipedia/HardyLittlewood/#HardyLittlewood___Nat___primeTupleCounting","anchor":"HardyLittlewood___Nat___primeTupleCounting","docHtml":"\n For a given tuple $(m_1, \\dots, m_k)$, this counts number of admissible\nprime constellations $(p, p + m_1, \\dots, p + m_k)$ where $p \\leq n$.
"},"HardyLittlewood.FirstHardyLittlewoodConjectureFor":{"url":"/FormalConjectures/Wikipedia/HardyLittlewood/#HardyLittlewood___FirstHardyLittlewoodConjectureFor","anchor":"HardyLittlewood___FirstHardyLittlewoodConjectureFor"},"HardyLittlewood.first_hardy_littlewood_conjecture":{"url":"/FormalConjectures/Wikipedia/HardyLittlewood/#HardyLittlewood___first_hardy_littlewood_conjecture","anchor":"HardyLittlewood___first_hardy_littlewood_conjecture","docHtml":"\n Let $P = (m_1, \\dots, m_k)$ be a tuple of positive even integers. Let\n$\\pi_P(n)$ denote the number of primes $p\\leq n$ such that $(p, p + m_1, \\dots, p + m_k)$\nforms an admissible prime constellation. Let $w(q; m_1, \\dots, m_k)$ denote the\nnumber of distinct residues of $0, m_1, \\dots, m_k$ modulo $q$, and let\n$$\nC_P = 2 ^ k\\prod_{\\substack{q\\ \\text{prime} \\ q\\geq 3}}\n\\frac{1 - \\frac{w(q; m_1, \\dots, m_k)}{q}}{\\left(1 - \\frac{1}{q}\\right)^{k+1}}.\n$$\nThen\n$$\n\\pi_P(n)\\sim C_P\\int_2^n\\frac{dt}{\\log^{k+1}t}.\n$$
"},"HardyLittlewood.SecondHardyLittlewoodConjectureFor":{"url":"/FormalConjectures/Wikipedia/HardyLittlewood/#HardyLittlewood___SecondHardyLittlewoodConjectureFor","anchor":"HardyLittlewood___SecondHardyLittlewoodConjectureFor"},"HardyLittlewood.second_hardy_littlewood_conjecture":{"url":"/FormalConjectures/Wikipedia/HardyLittlewood/#HardyLittlewood___second_hardy_littlewood_conjecture","anchor":"HardyLittlewood___second_hardy_littlewood_conjecture","docHtml":"\n For integers $x, y \\geq 2$,\n$$\n\\pi(x + y) \\leq \\pi(x) + \\pi(y),\n$$\nwhere $\\pi(z)$ denotes the prime-counting function, giving the number of primes up to\nand including $z$.
"},"HardyLittlewood.not_first_and_secondHardyLittlewoodConjecture":{"url":"/FormalConjectures/Wikipedia/HardyLittlewood/#HardyLittlewood___not_first_and_secondHardyLittlewoodConjecture","anchor":"HardyLittlewood___not_first_and_secondHardyLittlewoodConjecture","docHtml":"\n Richards [Ri74] showed that only one of the two Hardy-Littlewood conjectures can be true.
\n\n [Ri74] Richards, Ian (1974).
\nClosedInvariantSubspace T is the type of non-trivial (different from H and {0}) closed\nsubspaces of a complex vector space H that are invariant under the action of linear map T.
\nClosedInvariantSubspace T is the type of non-trivial (different from H and {0}) closed\nsubspaces of a complex vector space H that are invariant under the action of linear map T.
\nClosedInvariantSubspace T is the type of non-trivial (different from H and {0}) closed\nsubspaces of a complex vector space H that are invariant under the action of linear map T.
\nClosedInvariantSubspace T is the type of non-trivial (different from H and {0}) closed\nsubspaces of a complex vector space H that are invariant under the action of linear map T.
\nClosedInvariantSubspace T is the type of non-trivial (different from H and {0}) closed\nsubspaces of a complex vector space H that are invariant under the action of linear map T.
\nClosedInvariantSubspace T is the type of non-trivial (different from H and {0}) closed\nsubspaces of a complex vector space H that are invariant under the action of linear map T.
\n Show that every bounded linear operator T : H → H on a separable Hilbert space H of dimension\nat least 2 has a non-trivial closed T-invariant subspace: a closed linear subspace W of H,\nwhich is different from H and from {0}, such that T ( W ) ⊂ W. One needs the assumption that\nthe dimension of H is at least 2 because otherwise any subspace would be either H or {0}.
\n Every (bounded) linear operator T : H → H on a finite-dimensional linear space H of dimension\nat least 2 has a non-trivial (closed) T-invariant subspace. This can be solved using the Jordan\nnormal form, which is\nnot yet in mathlib.
\n Every bounded linear operator T : H → H on a non-separable Hilbert space H has a\nnon-trivial closed T-invariant subspace. Such an invariant space is given by considering the\nclosure of the linear span of the orbit of any single non-zero vector.
\n Every normal linear operator T : H → H on a Hilbert space H of dimension at least 2 has a\nnon-trivial closed T-invariant subspace. If T is a multiple of the identity, one can take any\nnon-trivial subspace . If not, one can take any nontrivial spectral subspace of T.
\n There exists a bounded linear operator T on the l1 space (lp (fun (_ : ℕ) => ℂ) 1)) without\nnon-trivial closed T-invariant subspace Read 1985, see\nalso the first counterexample by Enflo Enflo 1987, submitted\nin 1981.
\n Consider the following operation on the natural numbers:\nIf the number is even, divide it by two.\nIf the number is odd, triple it and add one.
"},"CollatzConjecture.collatz_conjecture":{"url":"/FormalConjectures/Wikipedia/CollatzConjecture/#CollatzConjecture___collatz_conjecture","anchor":"CollatzConjecture___collatz_conjecture","docHtml":"\n Now form a sequence beginning with any positive integer, where each subsequent term is obtained\nby applying the operation defined above to the previous term.\nThe Collatz conjecture states that for any positive integer $n$, there exists a natural number\n$m$ such that the $m$-th term of the sequence is 1.
"},"LehmerTotient.lehmer_totient":{"url":"/FormalConjectures/Wikipedia/LehmerTotient/#LehmerTotient___lehmer_totient","anchor":"LehmerTotient___lehmer_totient","docHtml":"\n Does there exist a composite number $n > 1$ such that Euler’s totient function\n$\\varphi(n)$ divides $n - 1$?
"},"Conway99Graph.completeGraphIsClique":{"url":"/FormalConjectures/Wikipedia/Conway99Graph/#Conway99Graph___completeGraphIsClique","anchor":"Conway99Graph___completeGraphIsClique","docHtml":"\n A finset of vertices in a complete graph is always a clique.
"},"Conway99Graph.completeGraph_cliqueSet":{"url":"/FormalConjectures/Wikipedia/Conway99Graph/#Conway99Graph___completeGraph_cliqueSet","anchor":"Conway99Graph___completeGraph_cliqueSet","docHtml":"\n The only clique of size n in a complete graph on n vertices is the entire set of vertices.
\n Each two non-adjacent vertices have exactly two common neighbors.
"},"Conway99Graph.conway99Graph":{"url":"/FormalConjectures/Wikipedia/Conway99Graph/#Conway99Graph___conway99Graph","anchor":"Conway99Graph___conway99Graph","docHtml":"\n Does there exist an undirected graph with 99 vertices, in which each two adjacent vertices have\nexactly one common neighbor, and in which each two non-adjacent vertices have exactly two common\nneighbors?\nEquivalently, every edge should be part of a unique triangle and every non-adjacent pair should be\none of the two diagonals of a unique 4-cycle.\nThe first condition is equivalent to being locally linear.
"},"Conway99Graph.triangle_locallyLinear_and_nonEdgesAreDiagonals":{"url":"/FormalConjectures/Wikipedia/Conway99Graph/#Conway99Graph___triangle_locallyLinear_and_nonEdgesAreDiagonals","anchor":"Conway99Graph___triangle_locallyLinear_and_nonEdgesAreDiagonals","docHtml":"\n The triangle is an example with 3 vertices satisfying the condition.
"},"Conway99Graph.Conway9":{"url":"/FormalConjectures/Wikipedia/Conway99Graph/#Conway99Graph___Conway9","anchor":"Conway99Graph___Conway9","docHtml":"\n The box product of two triangles is an example with 9 vertices satisfying the condition.\n(This graph is the complement of the one described in https://vimeo.com/109815595\nand it is also isomorphic to it and to the Paley graph and the graph of the\n3-3 duoprism)
"},"Conway99Graph.conway9_nonEdgesAreDiagonals":{"url":"/FormalConjectures/Wikipedia/Conway99Graph/#Conway99Graph___conway9_nonEdgesAreDiagonals","anchor":"Conway99Graph___conway9_nonEdgesAreDiagonals"},"Conway99Graph.completeGraph_boxProd_completeGraph_cliqueSet":{"url":"/FormalConjectures/Wikipedia/Conway99Graph/#Conway99Graph___completeGraph_boxProd_completeGraph_cliqueSet","anchor":"Conway99Graph___completeGraph_boxProd_completeGraph_cliqueSet"},"Conway99Graph.conway9_locallyLinear":{"url":"/FormalConjectures/Wikipedia/Conway99Graph/#Conway99Graph___conway9_locallyLinear","anchor":"Conway99Graph___conway9_locallyLinear"},"MagicSquares.exists_magic_square_squares":{"url":"/FormalConjectures/Wikipedia/MagicSquares/#MagicSquares___exists_magic_square_squares","anchor":"MagicSquares___exists_magic_square_squares","docHtml":"\n Does there exist a $3 \\times 3$ matrix such that every entry is a distinct square,\nand all rows, columns, and diagonals add up to the same value?
\n\n 0 is excluded, as a Magic Square of Squares with 0 and 8 distinct squares is know is knownn.\nSee Magic Square of Squares
"},"MagicSquares.exists_semi_magic_square_cubes":{"url":"/FormalConjectures/Wikipedia/MagicSquares/#MagicSquares___exists_semi_magic_square_cubes","anchor":"MagicSquares___exists_semi_magic_square_cubes","docHtml":"\n Does there exist a $3 \\times 3$ semi-magic square whose entries are all distinct positive\ninteger cubes? A square is semi-magic if all rows and columns sum to the same total.
\n\n More precisely, we seek a $3 \\times 3$ matrix with entries $a_{ij}$ such that each\n$a_{ij} = n_{ij}^3$ for some positive integer $n_{ij}$, all nine cubes are distinct,\nand all row sums and column sums are equal.
\n\n
\nBunyakovsky conjecture\nIf a polynomial $f$ over integers satisfies both Schinzel and Bunyakovsky conditions,\nthere exist infinitely many natural numbers $m$ such that $f(m)$ is prime.
"},"GapConjecture.gap_conjecture":{"url":"/FormalConjectures/Wikipedia/GapConjecture/#GapConjecture___gap_conjecture","anchor":"GapConjecture___gap_conjecture","docHtml":"\n If a finitely generated group has superpolynomial growth, then with respect to any finite\ngenerating set its growth function is at least $e^{\\sqrt n}$ in Grigorchuk's preorder on\ngrowth functions, where the comparison is witnessed by linearly rescaling the radius.
"},"MoserWorm.Worms":{"url":"/FormalConjectures/Wikipedia/MoserWorm/#MoserWorm___Worms","anchor":"MoserWorm___Worms","docHtml":"\n The set of worms is the set of curves of length (at most) 1.\nWe formalize this as the set of ranges of 1-Lipschitz functions from [0,1] to ℝ².
\n The set of covers is the set of (measurable) sets\nthat cover every worm by translation and rotation.
"},"MoserWorm.disc_mem_worm_covers":{"url":"/FormalConjectures/Wikipedia/MoserWorm/#MoserWorm___disc_mem_worm_covers","anchor":"MoserWorm___disc_mem_worm_covers","docHtml":"\n A disc of radius 1 / 2 is a worm cover.
\n\n This follows by translating the center of the disc to the midpoint of the worm.
"},"MoserWorm.mosers_worm_problem":{"url":"/FormalConjectures/Wikipedia/MoserWorm/#MoserWorm___mosers_worm_problem","anchor":"MoserWorm___mosers_worm_problem","docHtml":"\nMoser's Worm Problem\nWhat is the minimal area (or greatest lower bound on the area)\nof a shape that can cover every unit-length curve?
"},"MoserWorm.mosers_worm_problem_upper_bound":{"url":"/FormalConjectures/Wikipedia/MoserWorm/#MoserWorm___mosers_worm_problem_upper_bound","anchor":"MoserWorm___mosers_worm_problem_upper_bound","docHtml":"\n There is a set of area 0.260437 that covers all worms.
\n\n
\nConvex Moser's Worm Problem\nWhat is the minimal area (or greatest lower bound on the area)\nof a
\n The minimal area of a convex shape that can cover every unit-length curve is attained.\nThis follows from the Blaschke selection theorem.
"},"MoserWorm.convex_mosers_worm_problem_upper_bound":{"url":"/FormalConjectures/Wikipedia/MoserWorm/#MoserWorm___convex_mosers_worm_problem_upper_bound","anchor":"MoserWorm___convex_mosers_worm_problem_upper_bound","docHtml":"\n There is a convex set of area 0.270911861 that covers all worms.
\n\n
\n 0.232239 is a lower bound on the area of a convex set that covers all worms.
\n\n
\n The Multibrot set of power n is the set of all parameters c : ℂ for which 0 does not\nescape to infinity under repeated application of z ↦ z ^ n + c.
\n The Mandelbrot set is the special case of the multibrot set for n = 2. In other words, it is the\nset of all parameters c : ℂ for which 0 does not escape to infinity under repeated application\nof z ↦ z ^ 2 + c.
\n The multibrotSet n is equivalently the set of all parameters c for which the orbit of 0\nunder z ↦ z ^ n + c does not leave the closed disk of radius 2 ^ (n - 1)⁻¹ around the origin.
\n The mandelbrot set is equivalently the set of all parameters c for which the orbit of 0\nunder z ↦ z ^ 2 + c does not leave the closed disk of radius two around the origin.
\n The MLC conjecture, stating that the mandelbrot set is locally connected.
"},"Mandelbrot.MLC_general_exponent":{"url":"/FormalConjectures/Wikipedia/Mandelbrot/#Mandelbrot___MLC_general_exponent","anchor":"Mandelbrot___MLC_general_exponent","docHtml":"\n A stronger version of the MLC conjecture, stating that all multibrots are locally connected.\nNote that we don't need to require 2 ≤ n because the conjecture holds in the trivial cases n = 0\nand n = 1 too.
\n We say that z : ℂ is part of an attracting cycle of period n of f : ℂ → ℂ if it is an\nn-periodic point (i.e. f^[n] z = z), f^[n] is differentiable at z, ‖deriv f^[n] z‖ is\nstrictly less than one, and n > 0.
\n For example, 0 is part of an attracting 2-cycle of z ↦ z ^ 2 - 1.
\n On the other hand, while 2 is part of a 1-cycle of z ↦ z ^ 2 - 2, that cycle is not\nattracting.
\n No function has an attracting cycle of period 0. This is important in that it means we don't\nneed to require 0 < n in the conjectures below.
\n The density of hyperbolicity conjecture, stating that the set of all parameters c for which\nfun z ↦ z ^ 2 + c has an attracting cycle is dense in the Mandelbrot set.
\n The density of hyperbolicity conjecture for Multibrot sets, stating that the set of all\nparameters c for which fun z ↦ z ^ n + c has an attracting cycle is dense in multibrotSet n.\nNote that we need to require 2 ≤ n because the conjecture is trivially false for n = 1.
\n The boundary of any Multibrot set is measurable because it is closed, so it makes sense to\nask about its area.
"},"Mandelbrot.volume_frontier_mandelbrotSet_eq_zero":{"url":"/FormalConjectures/Wikipedia/Mandelbrot/#Mandelbrot___volume_frontier_mandelbrotSet_eq_zero","anchor":"Mandelbrot___volume_frontier_mandelbrotSet_eq_zero","docHtml":"\n The boundary of the Mandelbrot set is conjectured to have zero area.
"},"Mandelbrot.volume_frontier_multibrotSet_eq_zero":{"url":"/FormalConjectures/Wikipedia/Mandelbrot/#Mandelbrot___volume_frontier_multibrotSet_eq_zero","anchor":"Mandelbrot___volume_frontier_multibrotSet_eq_zero","docHtml":"\n The boundary of any Multibrot set is conjectured to have zero area.\nNote that we don't need to exclude the trivial cases n = 0 and n = 1 because the conjecture\nholds for them.
\n
\n There are infinitely many real quadratic fields ℚ(√d) with class number one,\nwhere d > 1 is a squarefree integer.
\nStark–Heegner theorem : For any squarefree integer d < 0, the class number of the imaginary\nquadratic field Q(√d) is one if and only if d ∈ {-1, -2, -3, -7, -11, -19, -43, -67, -163}.
\n Say a subset I of a ring R is nilpotent if all its elements are nilpotent.
\n The R is the sum of all (two-sided) nil ideals of R.\nTags: Kothe Radical, upper nilradical
\n The Köthe conjecture: In any ring, the sum of two nil left ideals is nil.
"},"Koethe.KotherConjecture.variants.le_KotherRadical":{"url":"/FormalConjectures/Wikipedia/Koethe/#Koethe___KotherConjecture___variants___le_KotherRadical","anchor":"Koethe___KotherConjecture___variants___le_KotherRadical","docHtml":"\n The Köthe conjecture: every left nil radical is contained in the Köthe radical.
"},"Koethe.KotherConjecture.variants.general_matrix":{"url":"/FormalConjectures/Wikipedia/Koethe/#Koethe___KotherConjecture___variants___general_matrix","anchor":"Koethe___KotherConjecture___variants___general_matrix","docHtml":"\n The Köthe conjecture: for any nil ideal I of R, the matrix ideal M_n(I) is a nil ideal\nof the matrix ring M_n(R).
\n The Köthe conjecture: for any nil ideal I of R, the matrix ideal M_2(I) is a nil ideal\nof the matrix ring M_2(R).
\n The Köthe conjecture: for any positive integer n, the Köthe radical of R is the matrix ideal M_2(Nil*(R)).
\n The Amitsur Conjecture: If J is a nil ideal in R, then J[x] is a nil ideal of the polynomial ring R[x].\nThis is known to be false, see Agata Smoktunowicz,
\n The set of solutions to the Fermat-Catalan Conjecture, i.e. the\nset of solutions $(a,b,c,m,n,k)$ to the equation $a^m + b^n = c^k$\nwhere $\\frac 1 m + \\frac 1 n + \\frac 1 k < 1$.
"},"FermatCatalanConjecture.FermatCatalanSet":{"url":"/FormalConjectures/Wikipedia/FermatCatalanConjecture/#FermatCatalanConjecture___FermatCatalanSet","anchor":"FermatCatalanConjecture___FermatCatalanSet"},"FermatCatalanConjecture.fermatCatalanConjecture":{"url":"/FormalConjectures/Wikipedia/FermatCatalanConjecture/#FermatCatalanConjecture___fermatCatalanConjecture","anchor":"FermatCatalanConjecture___fermatCatalanConjecture","docHtml":"\n The proposition that the Fermat-Catalan Conjecture is true.
"},"FermatCatalanConjecture.fermat_catalan":{"url":"/FormalConjectures/Wikipedia/FermatCatalanConjecture/#FermatCatalanConjecture___fermat_catalan","anchor":"FermatCatalanConjecture___fermat_catalan","docHtml":"\n The Fermat–Catalan conjecture states that the equation\n$a^m + b^n = c^k$ has only finitely many solutions $(a,b,c,m,n,k)$ with distinct triplets of values\n$(a^m, b^n, c^k)$ where $a, b, c$ are positive coprime integers and $m, n, k$ are positive integers satisfying\n$\\frac 1 m + \\frac 1 n + \\frac 1 k < 1$.
"},"FermatCatalanConjecture.fermat_catalan.variants.darmon_granville":{"url":"/FormalConjectures/Wikipedia/FermatCatalanConjecture/#FermatCatalanConjecture___fermat_catalan___variants___darmon_granville","anchor":"FermatCatalanConjecture___fermat_catalan___variants___darmon_granville","docHtml":"\n By the Darmon-Granville theorem,\nfor any fixed choice of positive integers m, n and k satisfying $\\frac 1 m + \\frac 1 n + \\frac 1 k < 1$,\nonly finitely many coprime triples $(a, b, c)$ solving $a^m + b^n = c^k$ exist.
"},"Fermat.fermat_number_are_composite":{"url":"/FormalConjectures/Wikipedia/Fermat/#Fermat___fermat_number_are_composite","anchor":"Fermat___fermat_number_are_composite","docHtml":"\n Are Fermat numbers composite for all n > 4?
\n Are there infinitely many Fermat primes?
"},"Fermat.infinite_fermat_composite":{"url":"/FormalConjectures/Wikipedia/Fermat/#Fermat___infinite_fermat_composite","anchor":"Fermat___infinite_fermat_composite","docHtml":"\n Are there infinitely many composite Fermat numbers?
"},"Fermat.all_fermat_squarefree":{"url":"/FormalConjectures/Wikipedia/Fermat/#Fermat___all_fermat_squarefree","anchor":"Fermat___all_fermat_squarefree","docHtml":"\n Are all Fermat numbers are square-free?
"},"EllipticCurveRank.RatEllipticCurve":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___RatEllipticCurve","anchor":"EllipticCurveRank___RatEllipticCurve","docHtml":"\n A data structure representing isomoprhism classes of elliptic curves over ℚ.\nEvery elliptic curve over ℚ is isomorphic to one with Weierstrass equation y² = x³ + Ax + B,\nand the pair (A,B) is unique if it satisfy the reduced condition below.\nSee Section 5.1 in [PPVW2016].
\n A data structure representing isomoprhism classes of elliptic curves over ℚ.\nEvery elliptic curve over ℚ is isomorphic to one with Weierstrass equation y² = x³ + Ax + B,\nand the pair (A,B) is unique if it satisfy the reduced condition below.\nSee Section 5.1 in [PPVW2016].
\n A data structure representing isomoprhism classes of elliptic curves over ℚ.\nEvery elliptic curve over ℚ is isomorphic to one with Weierstrass equation y² = x³ + Ax + B,\nand the pair (A,B) is unique if it satisfy the reduced condition below.\nSee Section 5.1 in [PPVW2016].
\n A data structure representing isomoprhism classes of elliptic curves over ℚ.\nEvery elliptic curve over ℚ is isomorphic to one with Weierstrass equation y² = x³ + Ax + B,\nand the pair (A,B) is unique if it satisfy the reduced condition below.\nSee Section 5.1 in [PPVW2016].
\n A data structure representing isomoprhism classes of elliptic curves over ℚ.\nEvery elliptic curve over ℚ is isomorphic to one with Weierstrass equation y² = x³ + Ax + B,\nand the pair (A,B) is unique if it satisfy the reduced condition below.\nSee Section 5.1 in [PPVW2016].
\n Convert the structure RatEllipticCurve to a Weierstrass curve.
\n The rank of an elliptic curve over ℚ.
"},"EllipticCurveRank.RatEllipticCurve.naiveHeight":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___RatEllipticCurve___naiveHeight","anchor":"EllipticCurveRank___RatEllipticCurve___naiveHeight","docHtml":"\n The naïve height of an elliptic curve over ℚ.
"},"EllipticCurveRank.RatEllipticCurve.heightLE":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___RatEllipticCurve___heightLE","anchor":"EllipticCurveRank___RatEllipticCurve___heightLE","docHtml":"\n The set of elliptic curves over ℚ with naïve height less than or equal to a given height.
"},"EllipticCurveRank.RatEllipticCurve.card_heightLE_div_pow_five_div_six_tensto":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___RatEllipticCurve___card_heightLE_div_pow_five_div_six_tensto","anchor":"EllipticCurveRank___RatEllipticCurve___card_heightLE_div_pow_five_div_six_tensto","docHtml":"\n Formula (5.1.1) of [PPVW2016]: The number of elliptic curves over ℚ with naïve height at most\nH is asymptotically 2^(4/3)*3^(-3/2)/ζ(10) * H^(5/6).
\n Conjecture by Goldfeld and Katz–Sarnak: if elliptic curves over ℚ are ordered by their\nheights, then 50% of the curves have rank 0 and 50% have rank 1.\nSee p. 28 of https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Bhargava.pdf.
"},"EllipticCurveRank.RatEllipticCurve.avg_rank_lt_0885":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___RatEllipticCurve___avg_rank_lt_0885","anchor":"EllipticCurveRank___RatEllipticCurve___avg_rank_lt_0885","docHtml":"\n Theorem 3 of [BS2013]:\nwhen elliptic curves over ℚ are ordered by height, their average rank is < .885.
"},"EllipticCurveRank.RatEllipticCurve._08375_le_density_rank_zero_one":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___RatEllipticCurve____08375_le_density_rank_zero_one","anchor":"EllipticCurveRank___RatEllipticCurve____08375_le_density_rank_zero_one","docHtml":"\n Theorem 4 of [BS2013]:\nwhen elliptic curves over ℚ are ordered by height, a density of at least 83.75% have\nrank 0 or 1.
"},"EllipticCurveRank.RatEllipticCurve._02062_le_density_rank_zero":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___RatEllipticCurve____02062_le_density_rank_zero","anchor":"EllipticCurveRank___RatEllipticCurve____02062_le_density_rank_zero","docHtml":"\n Theorem 5 of [BS2013]:\nwhen elliptic curves over ℚ are ordered by height, a density of at least 20.62% have rank 0.
"},"EllipticCurveRank.RatEllipticCurve.unbounded_rank_conjecture":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___RatEllipticCurve___unbounded_rank_conjecture","anchor":"EllipticCurveRank___RatEllipticCurve___unbounded_rank_conjecture","docHtml":"\n From [PPVW2016], Section 3.1: \"from the mid-1960s to the present,\nit seems that most experts conjectured unboundedness.\"
"},"EllipticCurveRank.RatEllipticCurve.finite_twentyone_lt_finrank":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___RatEllipticCurve___finite_twentyone_lt_finrank","anchor":"EllipticCurveRank___RatEllipticCurve___finite_twentyone_lt_finrank","docHtml":"\n From [PPVW2016], Section 8.2:\n\"Our heuristic predicts (a) All but finitely many E ∈ ℰ satisfy rk E(ℚ) ≤ 21\".\nIn other words, there are only finitely many elliptic curves over ℚ (up to isomorphism)\nwith rank greater than 21.\nNotice that this contradicts the previous conjecture.
"},"EllipticCurveRank.RatEllipticCurve.rank_height_count_asymptotic":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___RatEllipticCurve___rank_height_count_asymptotic","anchor":"EllipticCurveRank___RatEllipticCurve___rank_height_count_asymptotic","docHtml":"\n [PPVW2016] 8.2(b): for 1 ≤ r ≤ 20, the number of elliptic curves over ℚ with rank r and\nnaïve height at most H is asymptotically H ^ ((21 - r) / 24 + o(1)).\nNote: ℰ_H in 8.2(b) should be ℰ_{≤H}, see the statement of Theorem 7.3.3.\nWhen r = 1, the exponent is 20 / 24 = 5 / 6, which agrees with the exponent in\ncard_heightLE_div_pow_five_div_six_tensto and is consistent with\nhalf_rank_zero_and_half_rank_one.
\n [PPVW2016] 8.2(c): the number of elliptic curves over ℚ with rank ≥ 21 and naïve height\nat most H is asymptotically at most H ^ o(1).
\n The elliptic curve over ℚ of rank at least 29 found by Elkies and Klagsbrun in 2024.\nIt has rank exactly 29 assuming the generalized Riemann hypothesis.
"},"EllipticCurveRank.WeierstrassCurve.Δ_elkiesKlagsbrun29":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___WeierstrassCurve_______elkiesKlagsbrun29","anchor":"EllipticCurveRank___WeierstrassCurve_______elkiesKlagsbrun29","docHtml":"\n See https://mathoverflow.net/a/478050.
"},"EllipticCurveRank.WeierstrassCurve.twentynine_le_rank_elkiesKlagsbrun29":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___WeierstrassCurve___twentynine_le_rank_elkiesKlagsbrun29","anchor":"EllipticCurveRank___WeierstrassCurve___twentynine_le_rank_elkiesKlagsbrun29","docHtml":"\n The rank of the Elkies-Klagsbrun curve is at least 29.
"},"EllipticCurveRank.WeierstrassCurve.rank_elkiesKlagsbrun29":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___WeierstrassCurve___rank_elkiesKlagsbrun29","anchor":"EllipticCurveRank___WeierstrassCurve___rank_elkiesKlagsbrun29","docHtml":"\n The rank of the Elkies-Klagsbrun curve is exactly 29.
"},"EllipticCurveRank.WeierstrassCurve.elkies28":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___WeierstrassCurve___elkies28","anchor":"EllipticCurveRank___WeierstrassCurve___elkies28","docHtml":"\n The elliptic curve over ℚ of rank at least 28 found by Elkies in 2006.\nIt has rank exactly 28 assuming the generalized Riemann hypothesis.
"},"EllipticCurveRank.WeierstrassCurve.Δ_elkies28":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___WeierstrassCurve_______elkies28","anchor":"EllipticCurveRank___WeierstrassCurve_______elkies28","docHtml":"\n See https://mathoverflow.net/a/478050.
"},"EllipticCurveRank.WeierstrassCurve.twentyeight_le_rank_elkies28":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___WeierstrassCurve___twentyeight_le_rank_elkies28","anchor":"EllipticCurveRank___WeierstrassCurve___twentyeight_le_rank_elkies28","docHtml":"\n The rank of the Elkies curve is at least 28.
"},"EllipticCurveRank.WeierstrassCurve.rank_elkies28":{"url":"/FormalConjectures/Wikipedia/EllipticCurveRank/#EllipticCurveRank___WeierstrassCurve___rank_elkies28","anchor":"EllipticCurveRank___WeierstrassCurve___rank_elkies28","docHtml":"\n The rank of the Elkies curve is exactly 28.
"},"EuclidNumbers.Euclid":{"url":"/FormalConjectures/Wikipedia/Euclid/#EuclidNumbers___Euclid","anchor":"EuclidNumbers___Euclid","docHtml":"\n The n-th Euclid number is the product of the first n prime numbers plus one.
"},"EuclidNumbers.infinite_prime_euclid_numbers":{"url":"/FormalConjectures/Wikipedia/Euclid/#EuclidNumbers___infinite_prime_euclid_numbers","anchor":"EuclidNumbers___infinite_prime_euclid_numbers","docHtml":"\n It is not known whether there is an inifinite number of prime Euclid numbers.
"},"EuclidNumbers.euclid_numbers_are_square_free":{"url":"/FormalConjectures/Wikipedia/Euclid/#EuclidNumbers___euclid_numbers_are_square_free","anchor":"EuclidNumbers___euclid_numbers_are_square_free","docHtml":"\n It is not known whether every Euclid number is a square-free number.
"},"FlintCooksonHills.flint_hills_series_converges":{"url":"/FormalConjectures/Wikipedia/FlintCooksonHills/#FlintCooksonHills___flint_hills_series_converges","anchor":"FlintCooksonHills___flint_hills_series_converges","docHtml":"\n The Flint Hills series summing $csc(n)^2 / n^3$ from $n=1$ to $\\infty$ converges.\n(Note that we 0-index the series below.)
"},"FlintCooksonHills.cookson_hills_series_converges":{"url":"/FormalConjectures/Wikipedia/FlintCooksonHills/#FlintCooksonHills___cookson_hills_series_converges","anchor":"FlintCooksonHills___cookson_hills_series_converges","docHtml":"\n The Cookson Hills series summing $sec(n)^2 / n^3$ from $n=1$ to $\\infty$ converges.
"},"Catalan.catalans_conjecture":{"url":"/FormalConjectures/Wikipedia/Catalan/#Catalan___catalans_conjecture","anchor":"Catalan___catalans_conjecture","docHtml":"\n The only natural number solution to the equation $x^a - y^b = 1$ such that $a, b > 1$ and\n$x, y > 0$ is given by $a = 2$, $b = 3$, $x = 3$, and $y = 2$.
"},"Catalan.pillais_conjecture":{"url":"/FormalConjectures/Wikipedia/Catalan/#Catalan___pillais_conjecture","anchor":"Catalan___pillais_conjecture","docHtml":"\n For positive integers a, b, and c, there are only finitely many positive solutions (x, y, m, n) to the\nequation $ax^n - by^m = c$ where $(m, n) \\neq (2, 2)$ and $x, y > 1$.
"},"LebesgueNagell.lebesgue_nagell":{"url":"/FormalConjectures/Wikipedia/Catalan/#LebesgueNagell___lebesgue_nagell","anchor":"LebesgueNagell___lebesgue_nagell","docHtml":"\nLebesgue-Nagell Equation Conjecture
\n\n For any odd prime $p$, the only integer solutions $(x, y)$ to the equation $x^2 - 2 = y^p$\nare $(x, y) = (\\pm 1, -1)$.
\n\n
\n The pair $(1, -1)$ is a solution to $x^2 - 2 = y^p$ for any odd $p$.
"},"LebesgueNagell.lebesgue_nagell_solution_neg_one":{"url":"/FormalConjectures/Wikipedia/Catalan/#LebesgueNagell___lebesgue_nagell_solution_neg_one","anchor":"LebesgueNagell___lebesgue_nagell_solution_neg_one","docHtml":"\n The pair $(-1, -1)$ is a solution to $x^2 - 2 = y^p$ for any odd $p$.
"},"RegularPrimes.IsRegularPrime":{"url":"/FormalConjectures/Wikipedia/RegularPrimes/#RegularPrimes___IsRegularPrime","anchor":"RegularPrimes___IsRegularPrime","docHtml":"\n A natural prime number p is regular if p is coprime with the order of the class group\nof the p-th cyclotomic field.
\n The prime 37 is not a regular prime.
"},"RegularPrimes.regularPrimes":{"url":"/FormalConjectures/Wikipedia/RegularPrimes/#RegularPrimes___regularPrimes","anchor":"RegularPrimes___regularPrimes","docHtml":"\n The set of regular primes.
"},"RegularPrimes.irregularPrimes":{"url":"/FormalConjectures/Wikipedia/RegularPrimes/#RegularPrimes___irregularPrimes","anchor":"RegularPrimes___irregularPrimes","docHtml":"\n The set of irregular primes.
"},"RegularPrimes.small_regular_primes":{"url":"/FormalConjectures/Wikipedia/RegularPrimes/#RegularPrimes___small_regular_primes","anchor":"RegularPrimes___small_regular_primes","docHtml":"\n The primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31 are regular.
"},"RegularPrimes.not_isRegularPrime_37_second":{"url":"/FormalConjectures/Wikipedia/RegularPrimes/#RegularPrimes___not_isRegularPrime_37_second","anchor":"RegularPrimes___not_isRegularPrime_37_second","docHtml":"\n The prime 37 is not a regular prime.
"},"RegularPrimes.isRegularPrime_iff_Bernoulli":{"url":"/FormalConjectures/Wikipedia/RegularPrimes/#RegularPrimes___isRegularPrime_iff_Bernoulli","anchor":"RegularPrimes___isRegularPrime_iff_Bernoulli","docHtml":"\n An equivanlent definitions of regualr prime p is that it does not divide the numerator of the\nfirst p-3 Bernoulli numbers. Not in Mathlib.
\n The set of irregular primes is infinite.
"},"RegularPrimes.RegularPrimeConjecture":{"url":"/FormalConjectures/Wikipedia/RegularPrimes/#RegularPrimes___RegularPrimeConjecture","anchor":"RegularPrimes___RegularPrimeConjecture","docHtml":"\n Conjecture: The set of regular primes is infinite.
"},"RegularPrimes.regularprime_conjecture":{"url":"/FormalConjectures/Wikipedia/RegularPrimes/#RegularPrimes___regularprime_conjecture","anchor":"RegularPrimes___regularprime_conjecture","docHtml":"\n Conjecture: The set of regular primes is infinite.
"},"Firoozbakht.firoozbakhtSeq":{"url":"/FormalConjectures/Wikipedia/Firoozbakht/#Firoozbakht___firoozbakhtSeq","anchor":"Firoozbakht___firoozbakhtSeq","docHtml":"\n The sequence of $\\sqrt[n]{p_n}$ where $p_n$ is the n:th prime number.
"},"Firoozbakht.firoozbakht_conjecture":{"url":"/FormalConjectures/Wikipedia/Firoozbakht/#Firoozbakht___firoozbakht_conjecture","anchor":"Firoozbakht___firoozbakht_conjecture","docHtml":"\nFiroozbakht's conjecture\nThe inequality $\\sqrt[n+1]{p_{n+1}} < \\sqrt[n]{p_n}$ holds for all prime numbers $p_n$.
"},"Firoozbakht.firoozbakht_conjecture_consequence":{"url":"/FormalConjectures/Wikipedia/Firoozbakht/#Firoozbakht___firoozbakht_conjecture_consequence","anchor":"Firoozbakht___firoozbakht_conjecture_consequence","docHtml":"\n The inequality $p_{n+1}-p_n < (\\log p_n)^2-\\log p_n$ holds for all $n>4$.\nA consequence of Firuzbakht's conjecture.
"},"SquarePacking.Square":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___Square","anchor":"SquarePacking___Square","docHtml":"\n A square of a particular side length as a subset of the Euclidean plane.\nNot including border, so that squares that touch at the border are disjoint,\nbut a square internal to another shape is a subset of that shape.
"},"SquarePacking.UnitSquare":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___UnitSquare","anchor":"SquarePacking___UnitSquare","docHtml":"\n The unit square as a subset of the Euclidean plane.
"},"SquarePacking.Circle":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___Circle","anchor":"SquarePacking___Circle","docHtml":"\n A circle of a particular radius as a subset of the Euclidean plane.\nNot including border, so that circles that touch at the border are disjoint,\nbut a circle internal to another shape is a subset of that shape.
"},"SquarePacking.UnitCircle":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___UnitCircle","anchor":"SquarePacking___UnitCircle","docHtml":"\n The unit circle as a subset of the Euclidean plane.
"},"SquarePacking.Packing":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___Packing","anchor":"SquarePacking___Packing","docHtml":"\n A structure representing a packing of n isometric embeddings\nof a set s inside a (presumably larger) set S.
\n The isometric equivalences\nthat represent the transformations of the base shape to their locations in the packing.
"},"SquarePacking.Packing.disjoint":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___Packing___disjoint","anchor":"SquarePacking___Packing___disjoint","docHtml":"\n The images of the embeddings are pairwise disjoint
"},"SquarePacking.Packing.inside":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___Packing___inside","anchor":"SquarePacking___Packing___inside","docHtml":"\n The images of the embeddings are all inside the larger set S
\n Eleven unit squares can be packed into a square of side length < 3.877084.
\n\n Reference: Wikipedia
"},"SquarePacking.least_eleven_square_packing_in_square":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___least_eleven_square_packing_in_square","anchor":"SquarePacking___least_eleven_square_packing_in_square","docHtml":"\n What is the smallest square that can contain 11 unit squares?
\n\n Reference: Wikipedia
"},"SquarePacking.seventeen_square_packing_in_square_bound":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___seventeen_square_packing_in_square_bound","anchor":"SquarePacking___seventeen_square_packing_in_square_bound","docHtml":"\n Seventeen unit squares can be packed into a square of side length < 4.6756.
\n\n Reference: Wikipedia
"},"SquarePacking.least_seventeen_square_packing_in_square":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___least_seventeen_square_packing_in_square","anchor":"SquarePacking___least_seventeen_square_packing_in_square","docHtml":"\n What is the smallest square that can contain 17 unit squares?
\n\n Reference: Wikipedia
"},"SquarePacking.three_square_packing_in_circle_bound":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___three_square_packing_in_circle_bound","anchor":"SquarePacking___three_square_packing_in_circle_bound","docHtml":"\n Three unit squares can be packed into a circle of radius $(5 \\sqrt{17}) / 16 \\approx 1.288$.
\n\n Reference: Wikipedia
"},"SquarePacking.least_three_square_packing_in_circle":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___least_three_square_packing_in_circle","anchor":"SquarePacking___least_three_square_packing_in_circle","docHtml":"\n What is the smallest circle that can contain 3 unit squares?
\n\n Reference: Wikipedia
"},"SquarePacking.twenty_one_circle_packing_in_square_bound":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___twenty_one_circle_packing_in_square_bound","anchor":"SquarePacking___twenty_one_circle_packing_in_square_bound","docHtml":"\n Twenty-one unit circles can be packed into a square of side length < 9.359.
\n\n Reference: Visualizations
"},"SquarePacking.least_twenty_one_circle_packing_in_square":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___least_twenty_one_circle_packing_in_square","anchor":"SquarePacking___least_twenty_one_circle_packing_in_square","docHtml":"\n What is the smallest square that can contain 21 unit circles?
"},"SquarePacking.fifteen_circle_packing_in_circle_bound":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___fifteen_circle_packing_in_circle_bound","anchor":"SquarePacking___fifteen_circle_packing_in_circle_bound","docHtml":"\n Fifteen unit circles can be packed into a circle of radius\n$1 + \\sqrt{6 + 2/\\sqrt{5} + 4 \\sqrt{1 + 2/\\sqrt{5}}} \\approx 4.521$.
\n\n Reference:\nGraham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ.\nDense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.
"},"SquarePacking.least_fifteen_circle_packing_in_circle":{"url":"/FormalConjectures/Wikipedia/SquarePacking/#SquarePacking___least_fifteen_circle_packing_in_circle","anchor":"SquarePacking___least_fifteen_circle_packing_in_circle","docHtml":"\n What is the smallest circle that can contain 15 unit circles?
\n\n Reference:\nGraham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ.\nDense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.
"},"EulerBrick.IsEulerBrick":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___IsEulerBrick","anchor":"EulerBrick___IsEulerBrick","docHtml":"\n An Euler brick is a rectangular cuboid where all edges and face diagonals have integer lengths.
"},"EulerBrick.IsPerfectCuboid":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___IsPerfectCuboid","anchor":"EulerBrick___IsPerfectCuboid","docHtml":"\n A perfect cuboid is an Euler brick with an integer space diagonal.
"},"EulerBrick.IsEulerHyperBrick":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___IsEulerHyperBrick","anchor":"EulerBrick___IsEulerHyperBrick","docHtml":"\n Generalization of an Euler brick to $n$-dimensional space.
"},"EulerBrick.perfect_euler_brick_existence":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___perfect_euler_brick_existence","anchor":"EulerBrick___perfect_euler_brick_existence","docHtml":"\n Is there a perfect Euler brick?
"},"EulerBrick.four_dim_euler_brick_existence":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___four_dim_euler_brick_existence","anchor":"EulerBrick___four_dim_euler_brick_existence","docHtml":"\n Is there an Euler brick in $4$-dimensional space?
"},"EulerBrick.n_dim_euler_brick_existence":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___n_dim_euler_brick_existence","anchor":"EulerBrick___n_dim_euler_brick_existence","docHtml":"\n Is there an Euler brick in $n$-dimensional space for any $n > 3$?
"},"EulerBrick.CuboidOneFor":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___CuboidOneFor","anchor":"EulerBrick___CuboidOneFor","docHtml":"\n Pairs of natural numbers for which the first Cuboid polynomial is irreducible.
"},"EulerBrick.CuboidOne":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___CuboidOne","anchor":"EulerBrick___CuboidOne","docHtml":"\n
\n The first Cuboid conjecture
\n\n The DeepMind prover agent has found a formal disproof of this statement.
\n\n An (independent) informal solution can be found here:\n
\n Pairs of natural numbers for which the second Cuboid polynomial is irreducible.
"},"EulerBrick.CuboidTwo":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___CuboidTwo","anchor":"EulerBrick___CuboidTwo","docHtml":"\n
\n The second Cuboid conjecture
"},"EulerBrick.CuboidThreeFor":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___CuboidThreeFor","anchor":"EulerBrick___CuboidThreeFor","docHtml":"\n Triplets of natural numbers for which the third Cuboid polynomial is irreducible.
"},"EulerBrick.CuboidThree":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___CuboidThree","anchor":"EulerBrick___CuboidThree","docHtml":"\n
\n The third Cuboid conjecture
"},"EulerBrick.cuboid_perfect_euler_brick":{"url":"/FormalConjectures/Wikipedia/EulerBrick/#EulerBrick___cuboid_perfect_euler_brick","anchor":"EulerBrick___cuboid_perfect_euler_brick","docHtml":"\n In [Sh12], Ruslan notes that a perfect Euler brick does not exist\nif all three Cuboid conjectures hold.
"},"Andrica.andrica_conjecture":{"url":"/FormalConjectures/Wikipedia/Andrica/#Andrica___andrica_conjecture","anchor":"Andrica___andrica_conjecture","docHtml":"\nAndrica's conjecture\nThe inequality $\\sqrt{p_{n+1}}-\\sqrt{p_n} < 1$ holds for all $n$, where $p_n$ is the $n$-th prime number.
"},"Andrica.andrica_conjecture.ferreira_large_n":{"url":"/FormalConjectures/Wikipedia/Andrica/#Andrica___andrica_conjecture___ferreira_large_n","anchor":"Andrica___andrica_conjecture___ferreira_large_n","docHtml":"\n Ferreira proved that Andrica's conjecture is true for sufficiently large n.
"},"Kourovka.«19.25».kourovka.«19.25»":{"url":"/FormalConjectures/Kourovka/«19_25»/#Kourovka____FLQQ_19___25_FLQQ____kourovka____FLQQ_19___25_FLQQ_","anchor":"Kourovka____FLQQ_19___25_FLQQ____kourovka____FLQQ_19___25_FLQQ_","docHtml":"\n Let $G$ and $H$ be finite groups of the same order with\n$\\sum_{g \\in G} \\phi(|g|) = \\sum_{h \\in H} \\phi(|h|)$,\nwhere $\\phi$ is the Euler totient function. Suppose that $G$ is simple. Is\n$H$ necessarily simple?
"},"Kourovka.«20.76».kourovka.«20.76»":{"url":"/FormalConjectures/Kourovka/«20_76»/#Kourovka____FLQQ_20___76_FLQQ____kourovka____FLQQ_20___76_FLQQ_","anchor":"Kourovka____FLQQ_20___76_FLQQ____kourovka____FLQQ_20___76_FLQQ_","docHtml":"\n Let $G$ be a finite $p$-group and assume that all abelian normal subgroups of $G$\nhave order at most $p^k$. Is it true that every abelian subgroup of $G$ has order at most\n$p^{2k}$?
"},"Constant1a.C1a":{"url":"/FormalConjectures/OptimizationConstants/«1a»/#Constant1a___C1a","anchor":"Constant1a___C1a","docHtml":"\nTao's Optimization constant 1a / An autocorrelation constant related to Sidon sets:\nThe biggest real number satisfying a certain inequality about (auto)convolutions\nand $L^2$-norms of functions.\nThis number is related to the maximal size of Sidon sets in additive combinatorics.
"},"Constant1a.c1a_lower_bound":{"url":"/FormalConjectures/OptimizationConstants/«1a»/#Constant1a___c1a_lower_bound","anchor":"Constant1a___c1a_lower_bound","docHtml":"\n The best known lower bound, proven by Matolcsi-Vinuesa in [M2010]
"},"Constant1a.c1a_upper_bound":{"url":"/FormalConjectures/OptimizationConstants/«1a»/#Constant1a___c1a_upper_bound","anchor":"Constant1a___c1a_upper_bound","docHtml":"\n The best known upper bound, proven by Yuksekgonul et al. in [Y2026]
"},"Constant1a.mem_Ico_c1a":{"url":"/FormalConjectures/OptimizationConstants/«1a»/#Constant1a___mem_Ico_c1a","anchor":"Constant1a___mem_Ico_c1a","docHtml":"\n How can the upper bound be improved?
"},"Constant1a.mem_Ioc_c1a":{"url":"/FormalConjectures/OptimizationConstants/«1a»/#Constant1a___mem_Ioc_c1a","anchor":"Constant1a___mem_Ioc_c1a","docHtml":"\n How can the lower bound be improved?
"},"Constant1a.c1a_eq":{"url":"/FormalConjectures/OptimizationConstants/«1a»/#Constant1a___c1a_eq","anchor":"Constant1a___c1a_eq","docHtml":"\n What is the exact value of the constant?
"},"Green72.AllowedSet":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___AllowedSet","anchor":"Green72___AllowedSet","docHtml":"\n We say a subset of $[N]^2$ is allowed for some $k$ if it contains no $k$ points\nwhich lie on a common line.
"},"Green72.AllowedSet.is_bounded":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___AllowedSet___is_bounded","anchor":"Green72___AllowedSet___is_bounded","docHtml":"\n We say a subset of $[N]^2$ is allowed for some $k$ if it contains no $k$ points\nwhich lie on a common line.
"},"Green72.AllowedSet.not_collinear":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___AllowedSet___not_collinear","anchor":"Green72___AllowedSet___not_collinear","docHtml":"\n We say a subset of $[N]^2$ is allowed for some $k$ if it contains no $k$ points\nwhich lie on a common line.
"},"Green72.AllowedSetSize":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___AllowedSetSize","anchor":"Green72___AllowedSetSize","docHtml":"\n The maximal size of an allowed set
"},"Green72.allowedSetSize_le":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___allowedSetSize_le","anchor":"Green72___allowedSetSize_le","docHtml":"\n By the pigeon hole principle, the size of a subset of an $N \\times N$ grid such that no $k$\npoints lie on a line is bounded by $\\leq (k - 1) * N$ for $N \\geq k$.
"},"Green72.NoKInLineFor":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___NoKInLineFor","anchor":"Green72___NoKInLineFor","docHtml":"\n $N, k$ when the AllowedSetSize of $N$ for $k$ is $k * N$.
"},"Green72.NoKInLine":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___NoKInLine","anchor":"Green72___NoKInLine","docHtml":"\n The no-k-in-line problem:\nFor $N \\geq k$ and $k > 1$, the AllowedSetSize in $(k - 1) * N$, i. e. on an $N \\times N$ subset,\nthere is a set of $k * N$ points for which no $k$ lie on a line (and not such a set of bigger size).
"},"Green72.green_72":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___green_72","anchor":"Green72___green_72","docHtml":"\nGreen's Open Problem 72 / No-three-in-line problem:\nThe no-k-in-line conjecture holds for $k = 3$.
"},"Green72.no_three_in_line":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___no_three_in_line","anchor":"Green72___no_three_in_line"},"Green72.green_72.variants.eventually":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___green_72___variants___eventually","anchor":"Green72___green_72___variants___eventually","docHtml":"\n Does the no-three-in-line problem hold when $N$ is big enough?
"},"Green72.no_three_in_line_le":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___no_three_in_line_le","anchor":"Green72___no_three_in_line_le","docHtml":"\n For $N \\leq 60$, this has been verfied with computers.
"},"Green72.no_k_in_line_big":{"url":"/FormalConjectures/GreensOpenProblems/«72»/#Green72___no_k_in_line_big","anchor":"Green72___no_k_in_line_big","docHtml":"\n In [GK2025] Grebennikov and Kwan prove the no-k-in-line conjecture for $k > 10 ^ 37$.
"},"Green7.green_7.variants.positive_density":{"url":"/FormalConjectures/GreensOpenProblems/«7»/#Green7___green_7___variants___positive_density","anchor":"Green7___green_7___variants___positive_density","docHtml":"\n Does Ulam's sequence have positive density?
"},"Green62.green_62":{"url":"/FormalConjectures/GreensOpenProblems/«62»/#Green62___green_62","anchor":"Green62___green_62","docHtml":"\n Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$.\nIs every $x \\in {1, \\ldots, p-1}$ congruent to some product $a_1 a_2$ where $a_1, a_2 \\in A$?
"},"Green38.𝔽₇":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38_________","anchor":"Green38_________","docHtml":"\n The vector space $\\mathbb{F}_7^n$.
"},"Green38.IntersectsOnlyAtZero":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___IntersectsOnlyAtZero","anchor":"Green38___IntersectsOnlyAtZero","docHtml":"\n $A - A$ intersects ${-1, 0, 1}^n$ only at $0$.
"},"Green38.ValidCardinalities":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___ValidCardinalities","anchor":"Green38___ValidCardinalities","docHtml":"\n The set of cardinalities of all subsets A where A - A intersects {-1, 0, 1}^n only at 0.
"},"Green38.green_38.test_n1_lower":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___green_38___test_n1_lower","anchor":"Green38___green_38___test_n1_lower","docHtml":"\n {0, 2, 4} is a valid independent set in C_7, giving cardinality 3.
"},"Green38.LargestAdmissibleCardinality":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___LargestAdmissibleCardinality","anchor":"Green38___LargestAdmissibleCardinality","docHtml":"\n The largest subset $A \\subset \\mathbb{F}_7^n$ for which $A - A$ intersects ${-1, 0, 1}^n$ only\nat $0$.
"},"Green38.C₁":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___C___","anchor":"Green38___C___","docHtml":"\n The lower bound constant $C_1 \\approx 3.2578$ from [Po20].
"},"Green38.C₂":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___C___-","anchor":"Green38___C___-","docHtml":"\n The upper bound constant $C_2 \\approx 3.3177$ from [La79].
"},"Green38.green_38.lower":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___green_38___lower","anchor":"Green38___green_38___lower","docHtml":"\n Can we improve the lower bound?
"},"Green38.green_38.upper":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___green_38___upper","anchor":"Green38___green_38___upper","docHtml":"\n Can we improve the best upper bound?
"},"Green38.green_38.variants.best_lower":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___green_38___variants___best_lower","anchor":"Green38___green_38___variants___best_lower","docHtml":"\n The current best lower bound is $(C_1 - o(1))^n \\leqslant |A|$ where\n$C_1 = 367^{1/5} \\approx 3.2578$ [Po20, Section 9.1].
"},"Green38.green_38.variants.best_upper":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___green_38___variants___best_upper","anchor":"Green38___green_38___variants___best_upper","docHtml":"\n The current best upper bound is $|A| \\leqslant (C_2 + o(1))^n$ where\n$C_2 = \\frac{7 \\cos(\\pi/7)}{1 + \\cos(\\pi/7)} \\approx 3.3177$ [La79, Corollary 5].
"},"Green38.green_38.test_zero_mem_validCardinalities":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___green_38___test_zero_mem_validCardinalities","anchor":"Green38___green_38___test_zero_mem_validCardinalities","docHtml":"\n 0 is a valid cardinality, since the empty set vacuously satisfies the condition.
"},"Green38.green_38.test_bound_above":{"url":"/FormalConjectures/GreensOpenProblems/«38»/#Green38___green_38___test_bound_above","anchor":"Green38___green_38___test_bound_above","docHtml":"\n The set of valid cardinalities we take the supremum over is bounded above.
"},"Green85.green_85":{"url":"/FormalConjectures/GreensOpenProblems/«85»/#Green85___green_85","anchor":"Green85___green_85","docHtml":"\n Suppose that $A$ is an open subset of $[0, 1]^2$ with measure $\\alpha$. Are there four points in\n$A$ determining an axis-parallel rectangle with area $\\gt c \\alpha^2$?
"},"Green85.green_85_loose":{"url":"/FormalConjectures/GreensOpenProblems/«85»/#Green85___green_85_loose","anchor":"Green85___green_85_loose","docHtml":"\n From [Gr24] \"It is quite easy to show using Cauchy-Schwarz that there must be such a rectangle with\narea $\\gg \\alpha^2 (\\log 1/\\alpha)^{-1}$.\"
"},"Green37.IsAPCover":{"url":"/FormalConjectures/GreensOpenProblems/«37»/#Green37___IsAPCover","anchor":"Green37___IsAPCover","docHtml":"\nA contains an arithmetic progression of length k and common difference d for every d ∈ {1, …, N}.
\n The minimum size of a subset of ℕ that contains, for each d = 1, …, N,\nan arithmetic progression of length k with common difference d.
\n Given a natural number N, what is the smallest size of a subset of ℕ that contains, for each d = 1, …, N,\nan arithmetic progression of length k with common difference d.
\n Asymptotic version: determine the asymptotic behavior of m(N, k) as N grows.\nThe solver should determine what function f : ℕ → ℝ eventually equals (fun N ↦ (m N k : ℝ)).
\n Determine the asymptotic equivalence class (theta) of m(N, k).
\n Determine an upper bound (big O) for m(N, k).
\n Determine a strict upper bound (little o) for m(N, k).
\n Let $F(N)$ be the largest Sidon subset of $[N]$.
"},"Green31.green_31.lower":{"url":"/FormalConjectures/GreensOpenProblems/«31»/#Green31___green_31___lower","anchor":"Green31___green_31___lower","docHtml":"\n Can we improve the lower bound $N^{1/2} + O(1)$, at least for infinitely many $N$?
"},"Green31.green_31.variants.lower_eventually":{"url":"/FormalConjectures/GreensOpenProblems/«31»/#Green31___green_31___variants___lower_eventually","anchor":"Green31___green_31___variants___lower_eventually","docHtml":"\n Can we improve the lower bound $N^{1/2} + O(1)$, for all sufficiently large $N$?
"},"Green31.green_31.upper":{"url":"/FormalConjectures/GreensOpenProblems/«31»/#Green31___green_31___upper","anchor":"Green31___green_31___upper","docHtml":"\n Can we improve the upper bound $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25], at least for infinitely\nmany $N$?
"},"Green31.green_31.variants.upper_eventually":{"url":"/FormalConjectures/GreensOpenProblems/«31»/#Green31___green_31___variants___upper_eventually","anchor":"Green31___green_31___variants___upper_eventually","docHtml":"\n Can we improve the upper bound $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25], for all sufficiently\nlarge $N$?
"},"Green31.green_31.variants.upper_li69":{"url":"/FormalConjectures/GreensOpenProblems/«31»/#Green31___green_31___variants___upper_li69","anchor":"Green31___green_31___variants___upper_li69","docHtml":"\n [Li69] proved $F(n) \\le n^{1/2} + n^{1/4} + O(1)$.
"},"Green31.green_31.variants.upper_bfr23":{"url":"/FormalConjectures/GreensOpenProblems/«31»/#Green31___green_31___variants___upper_bfr23","anchor":"Green31___green_31___variants___upper_bfr23","docHtml":"\n [BFR23] obtained a small improvement, getting an upper bound of $F(N) \\le N^{1/2} + 0.998 N^{1/4}$\nfor large $N$.
"},"Green31.green_31.variants.upper_cho25":{"url":"/FormalConjectures/GreensOpenProblems/«31»/#Green31___green_31___variants___upper_cho25","anchor":"Green31___green_31___variants___upper_cho25","docHtml":"\n The upper bound was further improved to $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25].
"},"Green31.green_31.variants.zmod_p":{"url":"/FormalConjectures/GreensOpenProblems/«31»/#Green31___green_31___variants___zmod_p","anchor":"Green31___green_31___variants___zmod_p","docHtml":"\n It is not known whether or not there exists a Sidon subset of $\\mathbb{Z}/p\\mathbb{Z}$ of size\n$(1 + o(1))\\sqrt{p}$, for all $p$ [Gr24].
"},"Green31.green_31.variants.abelian":{"url":"/FormalConjectures/GreensOpenProblems/«31»/#Green31___green_31___variants___abelian","anchor":"Green31___green_31___variants___abelian","docHtml":"\n It is not known whether, if $G$ is an abelian group of size $n$, there always exists a Sidon subset\nof $G$ of size $0.01\\sqrt{n}$ [Gr24].
"},"Green31.green_31.variants.sidon_01n":{"url":"/FormalConjectures/GreensOpenProblems/«31»/#Green31___green_31___variants___sidon_01n","anchor":"Green31___green_31___variants___sidon_01n","docHtml":"\n Another very nice old problem is whether there is a Sidon subset of ${0, 1}^n$ of size $N^{0.51}$,\nwhere $N = 2^n$ [Gr24].
"},"Green31.green_31.variants.sidon_01n_clz01":{"url":"/FormalConjectures/GreensOpenProblems/«31»/#Green31___green_31___variants___sidon_01n_clz01","anchor":"Green31___green_31___variants___sidon_01n_clz01","docHtml":"\n The best-known upper bound for a Sidon subset of ${0, 1}^n$ ($N = 2^n$) is $N^{0.5753}$ [CLZ01].
"},"Green15.green_15":{"url":"/FormalConjectures/GreensOpenProblems/«15»/#Green15___green_15","anchor":"Green15___green_15","docHtml":"\n Does there exist a Lipschitz function $f : \\mathbb{N} \\to \\mathbb{Z}$ whose graph\n$\\Gamma = {(n, f(n)) : n \\in \\mathbb{N}} \\subseteq \\mathbb{Z}^2$ is free of 3-term progressions?
"},"Green15.green_15_ap4":{"url":"/FormalConjectures/GreensOpenProblems/«15»/#Green15___green_15_ap4","anchor":"Green15___green_15_ap4","docHtml":"\n The answer is YES for 4-term progressions [BJP14].
"},"Green9.r":{"url":"/FormalConjectures/GreensOpenProblems/«9»/#Green9___r","anchor":"Green9___r","docHtml":"\n The quantity $r_k(N)$, defined as the size of the largest subset of ${1, \\dots, N}$ without\nnon-trivial $k$-term arithmetic progressions.
"},"Green9.green_9_i":{"url":"/FormalConjectures/GreensOpenProblems/«9»/#Green9___green_9_i","anchor":"Green9___green_9_i","docHtml":"\n Problem 9 (i): is $r_3(N) \\ll N(\\log N)^{-10}$?
\n\n Solved in [BlSi20].
"},"Green9.green_9_ii":{"url":"/FormalConjectures/GreensOpenProblems/«9»/#Green9___green_9_ii","anchor":"Green9___green_9_ii","docHtml":"\n Problem 9 (ii): is $r_5(N) \\ll N(\\log N)^{-c}$?
"},"Green9.green_9_iii":{"url":"/FormalConjectures/GreensOpenProblems/«9»/#Green9___green_9_iii","anchor":"Green9___green_9_iii","docHtml":"\n Problem 9 (iii): is $r_4(\\mathbf{F}_5^n) \\ll N^{1-c}$, where $N=5^n$?
"},"Green26.𝔽":{"url":"/FormalConjectures/GreensOpenProblems/«26»/#Green26______","anchor":"Green26______","docHtml":"\n The vector space $\\mathbb{F}_p^n$.
"},"Green26.𝔽₃":{"url":"/FormalConjectures/GreensOpenProblems/«26»/#Green26_________","anchor":"Green26_________","docHtml":"\n The vector space $\\mathbb{F}_3^n$.
"},"Green26.StandardCube":{"url":"/FormalConjectures/GreensOpenProblems/«26»/#Green26___StandardCube","anchor":"Green26___StandardCube","docHtml":"\n The standard cube in $\\mathbb{F}_p^n$ is the set of points with coordinates in ${0, 1}$.
"},"Green26.IsCube":{"url":"/FormalConjectures/GreensOpenProblems/«26»/#Green26___IsCube","anchor":"Green26___IsCube","docHtml":"\n A cube is the image of $\\lbrace 0, 1\\rbrace^n$ under a linear automorphism.
"},"Green26.green_26":{"url":"/FormalConjectures/GreensOpenProblems/«26»/#Green26___green_26","anchor":"Green26___green_26","docHtml":"\n Let $A_1, \\dots, A_{100}$ be \"cubes\" in $\\mathbb{F}^n_3$.\nIs it true that $A_1 + \\dots + A_{100} = \\mathbb{F}^n_3$?
"},"Green26.green_26.variants.yu25":{"url":"/FormalConjectures/GreensOpenProblems/«26»/#Green26___green_26___variants___yu25","anchor":"Green26___green_26___variants___yu25","docHtml":"\n [Yu25] has solved the original problem (with 100 replaced by 4)
"},"Green26.green_26.variants.alm91":{"url":"/FormalConjectures/GreensOpenProblems/«26»/#Green26___green_26___variants___alm91","anchor":"Green26___green_26___variants___alm91","docHtml":"\n [ALM91] showed that if 100 is replaced by $\\leq c(p) \\log n$ then the result is true for\n$\\mathbb{F}^n_p$.
"},"Green26.green_26.variants.open":{"url":"/FormalConjectures/GreensOpenProblems/«26»/#Green26___green_26___variants___open","anchor":"Green26___green_26___variants___open","docHtml":"\n The analogous problem in $\\mathbb{F}^n_p$ remains open. [Gr24]
"},"Green57.tripleKernel":{"url":"/FormalConjectures/GreensOpenProblems/«57»/#Green57___tripleKernel","anchor":"Green57___tripleKernel","docHtml":"\n Uniform average over pairs (x₁, x₂) in G × G, with the third variable determined by\nthe relation x₁ + x₂ + x₃ = g. The functions fᵢ are complex-valued.
\n The generating family of functions for Φ(G). The functions fᵢ : G × G → ℂ are\nbounded by 1 in sup norm.
\n The generating family of functions for Φ′(G), where the third kernel depends only on\nx₁ + x₂.
\n The space Φ(G) of \"convex combinations\" of kernels from baseΦ.
\n Since baseΦ G is balanced (closed under multiplication by unit complex numbers) and\ncontains 0, the absolutely convex hull over ℂ (i.e., the set of\n$\\sum c_i \\phi_i$ with $\\phi_i \\in \\text{baseΦ}$ and $\\sum |c_i| \\le 1$)\ncoincides with the real convex hull.
\n The restricted space Φ′(G) of \"convex combinations\" of kernels from baseΦ'.
\n The linear functional on G → ℂ given by φ ↦ Re ∑ g, a g * φ g.
\n The support function of a set S with respect to a linear functional a:\nsup_{φ ∈ S} Re ⟨a, φ⟩.
\n For $G = \\mathbb{Z}/3\\mathbb{Z}$ and the functional $a(0) = -1$, $a(1) = -3$, $a(2) = 3$,\ndoes the support function of $\\Phi$ at $a$ strictly exceed that of $\\Phi'$?
\n\n Numerical evidence suggests the answer is yes:\n$$\\sup_{\\varphi \\in \\Phi'(\\mathbb{Z}/3\\mathbb{Z})} \\operatorname{Re}\\langle a, \\varphi \\rangle\n;<; \\sup_{\\varphi \\in \\Phi(\\mathbb{Z}/3\\mathbb{Z})} \\operatorname{Re}\\langle a, \\varphi \\rangle.$$
\n\n The DeepMind prover agent provided a formal proof, showing that $\\frac{183095}{30000}$ separates the\nsupport functions.
"},"Green57.green_57.variants.z3":{"url":"/FormalConjectures/GreensOpenProblems/«57»/#Green57___green_57___variants___z3","anchor":"Green57___green_57___variants___z3","docHtml":"\n Do $\\Phi(\\mathbb{Z}/3\\mathbb{Z})$ and $\\Phi'(\\mathbb{Z}/3\\mathbb{Z})$ coincide?
\n\n Numerical evidence suggests the answer is no: the integer functional $a = (-1, -3, 3)$\nseparates the two spaces. A branch-and-bound verification over the phase variables shows\n$\\max_{\\Phi'} \\operatorname{Re}\\langle a, \\varphi \\rangle < 6.112 < 6.115 \\approx\n\\max_{\\Phi} \\operatorname{Re}\\langle a, \\varphi \\rangle$.
"},"Green57.green_57":{"url":"/FormalConjectures/GreensOpenProblems/«57»/#Green57___green_57","anchor":"Green57___green_57","docHtml":"\n Is it true that for every finite abelian group $G$ the spaces $\\Phi(G)$ and $\\Phi'(G)$,\nobtained from kernels $\\phi(g) = \\mathbb{E}
\n Green guesses that the answer is probably 'no'.
"},"Green77.green_77":{"url":"/FormalConjectures/GreensOpenProblems/«77»/#Green77___green_77","anchor":"Green77___green_77","docHtml":"\n Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$\ndetermined by them?
"},"Green1.green_1":{"url":"/FormalConjectures/GreensOpenProblems/«1»/#Green1___green_1","anchor":"Green1___green_1","docHtml":"\n Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set\nof size at least $\\frac n 3 + Ω(n)$, where $Ω(n) → ∞$ as $n → ∞$?
"},"Green60.green_60":{"url":"/FormalConjectures/GreensOpenProblems/«60»/#Green60___green_60","anchor":"Green60___green_60","docHtml":"\n Is there an absolute constant $c > 0$ such that, whenever $A ⊆ \\mathbb{N}$ is a set of squares\nwith $|A| ≥ 2$, the sumset $A + A$ satisfies $|A + A| ≥ |A|^{1 + c}$?
"},"Green14.mixedMonoAPGuaranteeSet":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___mixedMonoAPGuaranteeSet","anchor":"Green14___mixedMonoAPGuaranteeSet","docHtml":"\n The set of natural numbers $N$ such that any 2-coloring of ${1, ..., N}$ contains a monochromatic\narithmetic progression of length $k$ (color 0) or length $r$ (color 1).
"},"Green14.W":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W","anchor":"Green14___W","docHtml":"\n We define the 2-colour van der Waerden numbers $W(k, r)$ to be the least quantities such that if\n${1, ... , W(k, r)}$ is coloured red and blue then there is either a red $k$-term progression\nor a blue $r$-term progression.
"},"Green14.green_14_polynomial":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___green_14_polynomial","anchor":"Green14___green_14_polynomial","docHtml":"\n Is $W(k, r)$ a polynomial in $r$, for fixed $k$?
\n\n We formulate this as asking if $W(k, r)$ has polynomial growth in $r$.\nWe know it is not the case for $k = 3$ [Gr21, p.3].
"},"Green14.green_14_polynomial_k_eq_3":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___green_14_polynomial_k_eq_3","anchor":"Green14___green_14_polynomial_k_eq_3","docHtml":"\n We know $W(3, r)$ does not have polynomial growth in $r$ [Gr21, p.3].
"},"Green14.green_14_quadratic":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___green_14_quadratic","anchor":"Green14___green_14_quadratic","docHtml":"\n Is $W(3, r) \\ll r^2$?
\n\n [Gr21] proves a superpolynomial lower bound $W(3, r) \\gg \\exp(c(\\log r)^{4/3-o(1)})$.
"},"Green14.green_14_lower_bound_green":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___green_14_lower_bound_green","anchor":"Green14___green_14_lower_bound_green","docHtml":"\n [Gr21] proved a lower bound of shape $W(3, r) \\gg \\exp(c(\\log r)^{4/3-o(1)})$.
"},"Green14.green_14_lower_bound_hunter":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___green_14_lower_bound_hunter","anchor":"Green14___green_14_lower_bound_hunter","docHtml":"\n [Hu22] improved this to $W(3, r) \\gg \\exp(c(\\log r)^{2-o(1)})$.
"},"Green14.green_14_lower_bound_brown_landman_robertson":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___green_14_lower_bound_brown_landman_robertson","anchor":"Green14___green_14_lower_bound_brown_landman_robertson","docHtml":"\n [BLR08] proved $W(3, r) \\gg r^{2 - 1/\\log \\log r}$.
"},"Green14.green_14_lower_bound_li_shu":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___green_14_lower_bound_li_shu","anchor":"Green14___green_14_lower_bound_li_shu","docHtml":"\n [LiSh10] proved $W(3, r) \\gg (r / \\log r)^2$.
"},"Green14.green_14_upper_bound_schoen":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___green_14_upper_bound_schoen","anchor":"Green14___green_14_upper_bound_schoen","docHtml":"\n [Sc20] proves the upper bound $W(3, r) < \\exp(r^{1-c})$ for some $c > 0$.
"},"Green14.green_14_upper_bound_kelley_meka":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___green_14_upper_bound_kelley_meka","anchor":"Green14___green_14_upper_bound_kelley_meka","docHtml":"\n [KeMe23] gives a corresponding upper bound $W(3, r) \\ll \\exp(C(\\log r)^C)$.
"},"Green14.green_14_variant_2r2":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___green_14_variant_2r2","anchor":"Green14___green_14_variant_2r2","docHtml":"\n It remains an interesting open problem to actually write down a colouring showing (say)\n$W(3, r) \\ge 2r^2$ for some $r$. [Gr24]
"},"Green14.W_3_3":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_3","anchor":"Green14___W_3_3","docHtml":"\n $W(3, 3) = 9$ from [AKS14].
"},"Green14.W_3_4":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_4","anchor":"Green14___W_3_4","docHtml":"\n $W(3, 4) = 18$ from [AKS14].
"},"Green14.W_3_5":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_5","anchor":"Green14___W_3_5","docHtml":"\n $W(3, 5) = 22$ from [AKS14].
"},"Green14.W_3_6":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_6","anchor":"Green14___W_3_6","docHtml":"\n $W(3, 6) = 32$ from [AKS14].
"},"Green14.W_3_7":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_7","anchor":"Green14___W_3_7","docHtml":"\n $W(3, 7) = 46$ from [AKS14].
"},"Green14.W_3_8":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_8","anchor":"Green14___W_3_8","docHtml":"\n $W(3, 8) = 58$ from [AKS14].
"},"Green14.W_3_9":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_9","anchor":"Green14___W_3_9","docHtml":"\n $W(3, 9) = 77$ from [AKS14].
"},"Green14.W_3_10":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_10","anchor":"Green14___W_3_10","docHtml":"\n $W(3, 10) = 97$ from [AKS14].
"},"Green14.W_3_11":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_11","anchor":"Green14___W_3_11","docHtml":"\n $W(3, 11) = 114$ from [AKS14].
"},"Green14.W_3_12":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_12","anchor":"Green14___W_3_12","docHtml":"\n $W(3, 12) = 135$ from [AKS14].
"},"Green14.W_3_13":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_13","anchor":"Green14___W_3_13","docHtml":"\n $W(3, 13) = 160$ from [AKS14].
"},"Green14.W_3_14":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_14","anchor":"Green14___W_3_14","docHtml":"\n $W(3, 14) = 186$ from [AKS14].
"},"Green14.W_3_15":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_15","anchor":"Green14___W_3_15","docHtml":"\n $W(3, 15) = 218$ from [AKS14].
"},"Green14.W_3_16":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_16","anchor":"Green14___W_3_16","docHtml":"\n $W(3, 16) = 238$ from [AKS14].
"},"Green14.W_3_17":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_17","anchor":"Green14___W_3_17","docHtml":"\n $W(3, 17) = 279$ from [AKS14].
"},"Green14.W_3_18":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_18","anchor":"Green14___W_3_18","docHtml":"\n $W(3, 18) = 312$ from [AKS14].
"},"Green14.W_3_19":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_19","anchor":"Green14___W_3_19","docHtml":"\n $W(3, 19) = 349$ from [AKS14].
"},"Green14.W_3_20_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_20_lower","anchor":"Green14___W_3_20_lower","docHtml":"\n $W(3, 20) \\ge 389$ from [AKS14, Table 2].
"},"Green14.W_3_21_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_21_lower","anchor":"Green14___W_3_21_lower","docHtml":"\n $W(3, 21) \\ge 416$ from [AKS14, Table 2].
"},"Green14.W_3_22_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_22_lower","anchor":"Green14___W_3_22_lower","docHtml":"\n $W(3, 22) \\ge 464$ from [AKS14, Table 2].
"},"Green14.W_3_23_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_23_lower","anchor":"Green14___W_3_23_lower","docHtml":"\n $W(3, 23) \\ge 516$ from [AKS14, Table 2].
"},"Green14.W_3_24_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_24_lower","anchor":"Green14___W_3_24_lower","docHtml":"\n $W(3, 24) \\ge 593$ from [AKS14, Table 2].
"},"Green14.W_3_25_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_25_lower","anchor":"Green14___W_3_25_lower","docHtml":"\n $W(3, 25) \\ge 656$ from [AKS14, Table 2].
"},"Green14.W_3_26_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_26_lower","anchor":"Green14___W_3_26_lower","docHtml":"\n $W(3, 26) \\ge 727$ from [AKS14, Table 2].
"},"Green14.W_3_27_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_27_lower","anchor":"Green14___W_3_27_lower","docHtml":"\n $W(3, 27) \\ge 770$ from [AKS14, Table 2].
"},"Green14.W_3_28_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_28_lower","anchor":"Green14___W_3_28_lower","docHtml":"\n $W(3, 28) \\ge 827$ from [AKS14, Table 2].
"},"Green14.W_3_29_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_29_lower","anchor":"Green14___W_3_29_lower","docHtml":"\n $W(3, 29) \\ge 868$ from [AKS14, Table 2].
"},"Green14.W_3_30_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_30_lower","anchor":"Green14___W_3_30_lower","docHtml":"\n $W(3, 30) \\ge 903$ from [AKS14, Table 2].
"},"Green14.W_3_31_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_31_lower","anchor":"Green14___W_3_31_lower","docHtml":"\n $W(3, 31) > 930$ from [AKS14, Table 3].
"},"Green14.W_3_32_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_32_lower","anchor":"Green14___W_3_32_lower","docHtml":"\n $W(3, 32) > 1006$ from [AKS14, Table 3].
"},"Green14.W_3_33_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_33_lower","anchor":"Green14___W_3_33_lower","docHtml":"\n $W(3, 33) > 1063$ from [AKS14, Table 3].
"},"Green14.W_3_34_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_34_lower","anchor":"Green14___W_3_34_lower","docHtml":"\n $W(3, 34) > 1143$ from [AKS14, Table 3].
"},"Green14.W_3_35_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_35_lower","anchor":"Green14___W_3_35_lower","docHtml":"\n $W(3, 35) > 1204$ from [AKS14, Table 3].
"},"Green14.W_3_36_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_36_lower","anchor":"Green14___W_3_36_lower","docHtml":"\n $W(3, 36) > 1257$ from [AKS14, Table 3].
"},"Green14.W_3_37_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_37_lower","anchor":"Green14___W_3_37_lower","docHtml":"\n $W(3, 37) > 1338$ from [AKS14, Table 3].
"},"Green14.W_3_38_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_38_lower","anchor":"Green14___W_3_38_lower","docHtml":"\n $W(3, 38) > 1378$ from [AKS14, Table 3].
"},"Green14.W_3_39_lower":{"url":"/FormalConjectures/GreensOpenProblems/«14»/#Green14___W_3_39_lower","anchor":"Green14___W_3_39_lower","docHtml":"\n $W(3, 39) > 1418$ from [AKS14, Table 3].
"},"Green33.green_33":{"url":"/FormalConjectures/GreensOpenProblems/«33»/#Green33___green_33","anchor":"Green33___green_33","docHtml":"\n Are there infinitely many $q$ for which there is a set $A \\subset \\mathbb{Z}/q\\mathbb{Z}$,\n$|A| = (\\sqrt{2} + o(1))q^{1/2}$, with $A + A = \\mathbb{Z}/q\\mathbb{Z}$? [Gr24]
"},"Green33.green_33.sanity_sq_bound":{"url":"/FormalConjectures/GreensOpenProblems/«33»/#Green33___green_33___sanity_sq_bound","anchor":"Green33___green_33___sanity_sq_bound","docHtml":"\n Trivial lower bound: if $A + A = \\mathbb{Z}/q\\mathbb{Z}$, then $|A|^2 \\geq q$,\nsince the sumset $A + A$ has at most $|A|^2$ elements.
"},"Green39.proportionCoverable":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___proportionCoverable","anchor":"Green39___proportionCoverable","docHtml":"\n The proportion of subsets of $\\mathbb{Z}/p\\mathbb{Z}$ of size $k$ that can cover\n$\\mathbb{Z}/p\\mathbb{Z}$ using at most $c$ translates.
\n\n If p = 0 or k > p, return 0 by convention.
"},"Green39.proportionCoverable_p_p_1":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___proportionCoverable_p_p_1","anchor":"Green39___proportionCoverable_p_p_1"},"Green39.proportionCoverable_t_0":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___proportionCoverable_t_0","anchor":"Green39___proportionCoverable_t_0"},"Green39.proportionCoverable_2_1_2":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___proportionCoverable_2_1_2","anchor":"Green39___proportionCoverable_2_1_2"},"Green39.proportionCoverable_3_1_2":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___proportionCoverable_3_1_2","anchor":"Green39___proportionCoverable_3_1_2"},"Green39.proportionCoverable_a_gt_p":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___proportionCoverable_a_gt_p","anchor":"Green39___proportionCoverable_a_gt_p"},"Green39.proportionCoverable_7_4_2":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___proportionCoverable_7_4_2","anchor":"Green39___proportionCoverable_7_4_2"},"Green39.proportionCoverable_11_3_4":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___proportionCoverable_11_3_4","anchor":"Green39___proportionCoverable_11_3_4"},"Green39.proportionCoverable_11_4_3":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___proportionCoverable_11_4_3","anchor":"Green39___proportionCoverable_11_4_3"},"Green39.green_39":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___green_39","anchor":"Green39___green_39","docHtml":"\n If $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ is random, $|A| = \\sqrt{p}$, can we almost surely cover\n$\\mathbb{Z}/p\\mathbb{Z}$ with $100\\sqrt{p}$ translates of $A$? [Gr24]
"},"Green39.green_39.variant_101":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___green_39___variant_101","anchor":"Green39___green_39___variant_101","docHtml":"\n \"I do not know how to answer this even with 100 replaced by 1.01.\" [Gr24]\"
"},"Green39.green_39.variant_theta":{"url":"/FormalConjectures/GreensOpenProblems/«39»/#Green39___green_39___variant_theta","anchor":"Green39___green_39___variant_theta","docHtml":"\n Similar questions are interesting with $\\sqrt{p}$ replaced by $p^\\theta$ for any $\\theta \\le 1/2$. [Gr24]
"},"Green4.ProdFree":{"url":"/FormalConjectures/GreensOpenProblems/«4»/#Green4___ProdFree","anchor":"Green4___ProdFree","docHtml":"\n A set in a monoid is product-free if there are no elements x, y, z in the set such that\nx * y = z.
\n What is the largest product-free set in the alternating group $A_n$?
"},"Green4.extremalFamily":{"url":"/FormalConjectures/GreensOpenProblems/«4»/#Green4___extremalFamily","anchor":"Green4___extremalFamily","docHtml":"\n Defines a family of subsets of $A_n$ where each permutation $\\pi$ in a subset obeys $\\pi(x)$\nand $\\forall v \\in I$, \\pi(v)\\notin I$ for a fixed $x$ and $I$. It is easy to demonstrate that such\na subset is product-free, because for any a,b,c in such a set, $(a*b) (x)=a(b(x))\\notin I$ but $c(x)\nin I$
"},"Green4.large_green_4":{"url":"/FormalConjectures/GreensOpenProblems/«4»/#Green4___large_green_4","anchor":"Green4___large_green_4","docHtml":"\n In the case of large n, the problem was solved in\nOn the largest product-free subsets of the alternating groups.\nSpecifically, this theorem formalizes the statement of theorem 1.1 in the mentioned paper
"},"Green58.green_58":{"url":"/FormalConjectures/GreensOpenProblems/«58»/#Green58___green_58","anchor":"Green58___green_58","docHtml":"\n Suppose $A, B ⊆ {1, \\dots, N}$ both have size at least $N^{0.49}$. Must the sumset $A + B$\ncontain a composite number?
"},"Green41.pyjamaSet":{"url":"/FormalConjectures/GreensOpenProblems/«41»/#Green41___pyjamaSet","anchor":"Green41___pyjamaSet","docHtml":"\n The pyjama set is the set of points in the complex plane whose real part is within $\\varepsilon$ of\nan integer.
"},"Green41.coveringCopies":{"url":"/FormalConjectures/GreensOpenProblems/«41»/#Green41___coveringCopies","anchor":"Green41___coveringCopies","docHtml":"\n The set of valid numbers of rotated copies of the pyjama set of width ε that cover the plane.
"},"Green41.minCopies":{"url":"/FormalConjectures/GreensOpenProblems/«41»/#Green41___minCopies","anchor":"Green41___minCopies","docHtml":"\n The minimal number of rotated copies of the pyjama set of width ε needed to cover the plane.
"},"Green41.minCopies_set_nonempty":{"url":"/FormalConjectures/GreensOpenProblems/«41»/#Green41___minCopies_set_nonempty","anchor":"Green41___minCopies_set_nonempty","docHtml":"\n [Ma15] proved that for any $\\varepsilon > 0$, finitely many rotations of the pyjama set of width\n$\\varepsilon$ cover the plane. This implies that the set we are taking the infimum over in minCopies\nis non-empty.
\n How many rotated (about the origin) copies of the 'pyjama set'\n$\\{(x, y) \\in \\mathbb{R}^2 : \\text{dist}(x, \\mathbb{Z}) \\leq \\varepsilon\\}$ are needed to cover\n$\\mathbb{R}^2$?
\n\n In particular, can one find a better bound than the best-known bound from [KrLe25]?
"},"Green41.green_41.variants.exists_better_bound":{"url":"/FormalConjectures/GreensOpenProblems/«41»/#Green41___green_41___variants___exists_better_bound","anchor":"Green41___green_41___variants___exists_better_bound","docHtml":"\n Is there a better bound than the best-known bound from [KrLe25]?\nThis is an existential version of the main problem that does not require providing the bound explicitly.
"},"Green41.green_41.variants.polynomial_bound":{"url":"/FormalConjectures/GreensOpenProblems/«41»/#Green41___green_41___variants___polynomial_bound","anchor":"Green41___green_41___variants___polynomial_bound","docHtml":"\n Is $\\varepsilon^{-C}$ rotations enough?
"},"Green41.green_41.variants.kravitz_leng":{"url":"/FormalConjectures/GreensOpenProblems/«41»/#Green41___green_41___variants___kravitz_leng","anchor":"Green41___green_41___variants___kravitz_leng","docHtml":"\n [KrLe25] have established the first quantitative bound, showing via an analysis of [Ma15]'s method\nthat $\\exp\\exp\\exp(\\varepsilon^{-C})$ rotations suffice.
"},"Green19.IsCorner":{"url":"/FormalConjectures/GreensOpenProblems/«19»/#Green19___IsCorner","anchor":"Green19___IsCorner","docHtml":"\n A corner in $A$ with common difference $d$ [FSS20].
"},"Green19.S":{"url":"/FormalConjectures/GreensOpenProblems/«19»/#Green19___S","anchor":"Green19___S","docHtml":"\n From [FSS20]: given $A \\subseteq G \\times G$ and $d \\in G$, let\n$$S_d(A) = \\lbrace (x, y) \\in G \\times G : (x, y), (x + d, y), (x, y + d) \\in A \\rbrace$$
"},"Green19.𝔽₂":{"url":"/FormalConjectures/GreensOpenProblems/«19»/#Green19_________","anchor":"Green19_________","docHtml":"\n The group $G = \\mathbb{F}_2^n = (Z/2Z)^n$.
"},"Green19.ValidExponent":{"url":"/FormalConjectures/GreensOpenProblems/«19»/#Green19___ValidExponent","anchor":"Green19___ValidExponent","docHtml":"\n True if the given exponent satisfies Green's conditions [Gr26].
"},"Green19.C":{"url":"/FormalConjectures/GreensOpenProblems/«19»/#Green19___C","anchor":"Green19___C","docHtml":"\n The infimum of all valid exponents [Gr26].
"},"Green19.green_19":{"url":"/FormalConjectures/GreensOpenProblems/«19»/#Green19___green_19","anchor":"Green19___green_19","docHtml":"\n What is $C$, the infimum of all exponents $c$ for which the following is true, uniformly for\n$0 < \\alpha < 1$? Suppose that $A \\subset \\mathbb{F}_2^n \\times \\mathbb{F}_2^n$ is a set of density\n$\\alpha$. Write $N := 2^n$. Then there is some $d \\neq 0$ such that $A$ contains $\\gg \\alpha^c N^2$\ncorners $(x,y), (x,y+d), (x+d,y)$.
\n\n This question has been resolved by [FSS20], showing that $C = 4$.
"},"Green19.green_19.lower":{"url":"/FormalConjectures/GreensOpenProblems/«19»/#Green19___green_19___lower","anchor":"Green19___green_19___lower","docHtml":"\n [Ma21] showed that $3.13 \\leq C$.
"},"Green19.green_19.upper":{"url":"/FormalConjectures/GreensOpenProblems/«19»/#Green19___green_19___upper","anchor":"Green19___green_19___upper","docHtml":"\n [Ma21] showed that $C \\leq 4$.
"},"Green27.m":{"url":"/FormalConjectures/GreensOpenProblems/«27»/#Green27___m","anchor":"Green27___m","docHtml":"\n This is $m(p)$ in [Be23]: the size of the smallest set $A \\subset \\mathbb{Z} / p\\mathbb{Z}$ (with\nat least two elements) for which no element in the sumset $A + A$ has a unique representation.
"},"Green27.primesAtTop":{"url":"/FormalConjectures/GreensOpenProblems/«27»/#Green27___primesAtTop","anchor":"Green27___primesAtTop","docHtml":"\natTop restricted to prime numbers.
\n Best-known lower bound [Be23, Theorem 3].
"},"Green27.upperBest":{"url":"/FormalConjectures/GreensOpenProblems/«27»/#Green27___upperBest","anchor":"Green27___upperBest","docHtml":"\n Best-known upper bound [Be23, Theorem 5].
"},"Green27.green_27.equivalent":{"url":"/FormalConjectures/GreensOpenProblems/«27»/#Green27___green_27___equivalent","anchor":"Green27___green_27___equivalent","docHtml":"\n What is the size of the smallest set $A \\subset \\mathbb{Z} / p\\mathbb{Z}$ (with at least two elements)\nfor which no element in the sumset $A + A$ has a unique representation?
"},"Green27.green_27.lower":{"url":"/FormalConjectures/GreensOpenProblems/«27»/#Green27___green_27___lower","anchor":"Green27___green_27___lower","docHtml":"\n Propose a better lower bound along primes.
"},"Green27.green_27.upper":{"url":"/FormalConjectures/GreensOpenProblems/«27»/#Green27___green_27___upper","anchor":"Green27___green_27___upper","docHtml":"\n Propose a better upper bound along primes.
"},"Green27.green_27.variants.lower_be23":{"url":"/FormalConjectures/GreensOpenProblems/«27»/#Green27___green_27___variants___lower_be23","anchor":"Green27___green_27___variants___lower_be23","docHtml":"\n We have $m(p) \\geq \\omega(p) \\log p$ for some function $\\omega(p)$ tending to infinity [Be23, Theorem 3].
"},"Green27.green_27.variants.upper_be23":{"url":"/FormalConjectures/GreensOpenProblems/«27»/#Green27___green_27___variants___upper_be23","anchor":"Green27___green_27___variants___upper_be23","docHtml":"\n Upper bound: $m(p) \\ll (\\log p)^2$ [Be23, Theorem 5].
"},"Green27.green_27.variants.previous_lower":{"url":"/FormalConjectures/GreensOpenProblems/«27»/#Green27___green_27___variants___previous_lower","anchor":"Green27___green_27___variants___previous_lower","docHtml":"\n Previous best-known lower bound $\\log p \\ll m(p)$ from [St76].
"},"Green27.green_27.variants.previous_upper":{"url":"/FormalConjectures/GreensOpenProblems/«27»/#Green27___green_27___variants___previous_upper","anchor":"Green27___green_27___variants___previous_upper","docHtml":"\n Previous best-known upper bound $m(p) \\ll \\sqrt{p}$ from [Be23].
"},"Green50.green_50":{"url":"/FormalConjectures/GreensOpenProblems/«50»/#Green50___green_50","anchor":"Green50___green_50","docHtml":"\n Let $A \\subset \\mathbb{F}_2^n$ be a set of density $\\alpha > 0$. Does $10A$ contain a coset\nof some subspace of dimension at least $n - O(\\log(1/\\alpha))$?
\n\n More precisely: does there exist an absolute constant $C > 0$ such that for all $n \\geq 1$ and all\nnonempty $A \\subseteq \\mathbb{F}_2^n$ with density $\\alpha > 0$, the sumset $10A$ contains a coset\nof some subspace of dimension at least $n - C \\log_2(1/\\alpha)$?
\n\n The sumset $10A$ is defined as ${a_1 + a_2 + \\cdots + a_{10} : a_i \\in A}$, using the pointwise\nscalar multiplication notation 10 • A where • denotes the iterated addition of a set.
\n Note: We model $\\mathbb{F}_2^n$ as Fin n → ZMod 2, which is an $n$-dimensional vector space\nover $\\mathbb{F}_2$.
\n The number of triples $(x, y, g)$ in $G^3$ such that $g \\neq e$, and $(x, y), (gx, y), (x, gy)$ are\nall in $A$. These are called \"naive corners\" by [Au16].
\n\n Note: the shortened formulation from [Gr26] does not mention $g \\neq e$, but this is the original\nstatement from [Au16], which ensure non-trivial corners. Note however that [Au16] use more\ngenerally compact groups and not just finite discrete groups.
"},"Green18.green_18":{"url":"/FormalConjectures/GreensOpenProblems/«18»/#Green18___green_18","anchor":"Green18___green_18","docHtml":"\n Suppose that $G$ is a finite group, and let $A \\subset G \\times G$ be a subset of density $\\alpha$.\nIs it true that there are $\\gg_\\alpha |G|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$\nall lie in $A$?
\n\n Note: A is taken as $\\alpha$-dense, i.e. $|A| \\ge \\alpha |G|^2$ [Au16, Question 2]
"},"Green18.numBmzCorners":{"url":"/FormalConjectures/GreensOpenProblems/«18»/#Green18___numBmzCorners","anchor":"Green18___numBmzCorners","docHtml":"\n The number of triples $(x, y, g)$ in $G^3$ such that $g \\neq e$, and $(x, y), (xg, y), (x, gy)$ are\nall in $A$. These are called \"BMZ corners\" by [Au16].
"},"Green18.green_18.bmz_corners":{"url":"/FormalConjectures/GreensOpenProblems/«18»/#Green18___green_18___bmz_corners","anchor":"Green18___green_18___bmz_corners","docHtml":"\n [So13] proved this is true for \"BMZ corners\". Follows from the proof of Theorem 2.1, p.1456-1457.
"},"Green35.IsUnitIntervalDensity":{"url":"/FormalConjectures/GreensOpenProblems/«35»/#Green35___IsUnitIntervalDensity","anchor":"Green35___IsUnitIntervalDensity","docHtml":"\n A nonnegative integrable function on $[0,1]$ with total integral $1$.
"},"Green35.c":{"url":"/FormalConjectures/GreensOpenProblems/«35»/#Green35___c","anchor":"Green35___c","docHtml":"\n The infimum of $|f \\ast f|_p$ over unit-interval densities.
"},"Green35.green_35.lower":{"url":"/FormalConjectures/GreensOpenProblems/«35»/#Green35___green_35___lower","anchor":"Green35___green_35___lower","docHtml":"\n Lower bound for $c(p)$ for $1 < p \\le \\infty$, improving the known value at $p = 2$ or $p = \\infty$.
"},"Green35.green_35.upper":{"url":"/FormalConjectures/GreensOpenProblems/«35»/#Green35___green_35___upper","anchor":"Green35___green_35___upper","docHtml":"\n Upper bound for $c(p)$ for $1 < p \\le \\infty$, improving the best-known value at $p = \\infty$.
"},"Green35.variants.c_2_lower":{"url":"/FormalConjectures/GreensOpenProblems/«35»/#Green35___variants___c_2_lower","anchor":"Green35___variants___c_2_lower","docHtml":"\n Lower bound for $c(2)$ from Green's first paper ([Gr01]); the constant is sqrt(4/7) (about 0.7559).
\n Best-known lower bound for $c(\\infty)$ due to Cloninger and Steinerberger ([CS17]).
"},"Green35.variants.c_inf_upper":{"url":"/FormalConjectures/GreensOpenProblems/«35»/#Green35___variants___c_inf_upper","anchor":"Green35___variants___c_inf_upper","docHtml":"\n Best-known upper bound for $c(\\infty)$ due to Matolcsi and Vinuesa ([MV10]).
"},"Green35.variants.c_inf_lower_young":{"url":"/FormalConjectures/GreensOpenProblems/«35»/#Green35___variants___c_inf_lower_young","anchor":"Green35___variants___c_inf_lower_young","docHtml":"\n A comparison bound from Young's inequality.
"},"Green94.green_94_outer_measure":{"url":"/FormalConjectures/GreensOpenProblems/«94»/#Green94___green_94_outer_measure","anchor":"Green94___green_94_outer_measure","docHtml":"\n Let A ⊂ R be a set of positive outer measure. Does $A$ contain an affine copy of {1, 1/2, 1/4, . . . }?
\n The answer is \"no\".
"},"Green94.green_94":{"url":"/FormalConjectures/GreensOpenProblems/«94»/#Green94___green_94","anchor":"Green94___green_94","docHtml":"\n Let A ⊂ R be a set of positive measure. Does $A$ contain an affine copy of {1, 1/2, 1/4, . . . }?
\n The vector space $\\mathbb{F}_2^n$.
"},"Green40.hammingBall":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___hammingBall","anchor":"Green40___hammingBall","docHtml":"\n The Hamming ball of radius $r$ in $\\mathbb{F}_2^n$.
"},"Green40.IsCoveringSubspace":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___IsCoveringSubspace","anchor":"Green40___IsCoveringSubspace","docHtml":"\n $V$ is a covering subspace of $\\mathbb{F}_2^n$ by $H(r)$ if $V + H(r) = \\mathbb{F}_2^n$.
"},"Green40.minDensity":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___minDensity","anchor":"Green40___minDensity","docHtml":"\n The minimal covering density over all covering subspaces for a given n and r.\nWe compute in ℝ≥0∞ (ENNReal) to gracefully handle any potential divergence.
\n Let $f(r)$ be the smallest constant such that there exists an infinite sequence of $n$'s together\nwith subspaces $V_n \\leq \\mathbb{F}_2^n$ with $V_n + H(r) = \\mathbb{F}_2^n$ and\n$|V_n| = \\left(f(r) + o(1)\\right) \\frac{2^n}{|H(r)|}$.
"},"Green40.green_40":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___green_40","anchor":"Green40___green_40","docHtml":"\n Does $f(r) \\to \\infty$? [Gr24]
"},"Green40.green_40.sanity_f_one":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___green_40___sanity_f_one","anchor":"Green40___green_40___sanity_f_one","docHtml":"\n The only value known is $f(1) = 1$, which follows from the existence of the Hamming code [Gr24].
"},"Green40.green_40.upper_bound":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___green_40___upper_bound","anchor":"Green40___green_40___upper_bound","docHtml":"\n $f(r) \\le r^r / r! \\sim e^r$ [Gr24].
"},"Green40.green_40.f_eq_one_for_all":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___green_40___f_eq_one_for_all","anchor":"Green40___green_40___f_eq_one_for_all","docHtml":"\n The possibility that f(r) = 1 for all r has not been ruled out [Gr24]
"},"Green40.green_40.f_two_eq_one":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___green_40___f_two_eq_one","anchor":"Green40___green_40___f_two_eq_one","docHtml":"\n It is not known whether f(2) = 1 [Gr24]
"},"Green40.green_40.upper_bound_f_two":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___green_40___upper_bound_f_two","anchor":"Green40___green_40___upper_bound_f_two","docHtml":"\n The best-known upper bound for $f(2)$ is $1.4238$ [CHL97].
"},"Green40.hammingBallFinset":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___hammingBallFinset","anchor":"Green40___hammingBallFinset"},"Green40.IsCoveringFinset":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___IsCoveringFinset","anchor":"Green40___IsCoveringFinset"},"Green40.minDensityFinset":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___minDensityFinset","anchor":"Green40___minDensityFinset"},"Green40.f_tilde":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___f_tilde","anchor":"Green40___f_tilde"},"Green40.green_40.variants.arbitrary_subsets":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___green_40___variants___arbitrary_subsets","anchor":"Green40___green_40___variants___arbitrary_subsets","docHtml":"\n Does $\\tilde{f}(r) \\to \\infty$? [Gr24]
"},"Green40.green_40.variants.arbitrary_subsets_sanity_f_tilde_two":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___green_40___variants___arbitrary_subsets_sanity_f_tilde_two","anchor":"Green40___green_40___variants___arbitrary_subsets_sanity_f_tilde_two","docHtml":"\n It is known that $\\tilde{f}(2) = 1$ [St94].
"},"Green40.green_40.f_tilde_le_f":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___green_40___f_tilde_le_f","anchor":"Green40___green_40___f_tilde_le_f","docHtml":"\n We evidently have $\\tilde{f}(r) \\le f(r)$ [Gr24].
"},"Green40.f_all":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___f_all","anchor":"Green40___f_all"},"Green40.green_40.variants.all_n":{"url":"/FormalConjectures/GreensOpenProblems/«40»/#Green40___green_40___variants___all_n","anchor":"Green40___green_40___variants___all_n","docHtml":"\n Does $f_{\\text{all}}(r) \\to \\infty$? [Gr24]
"},"Green61.green_61":{"url":"/FormalConjectures/GreensOpenProblems/«61»/#Green61___green_61","anchor":"Green61___green_61","docHtml":"\n Suppose that $A + A$ contains the first $n$ squares. Is $|A| \\geq n^{1 - o(1)}$?
\n\n It is known that necessarily $|A| \\geq n^{2/3 - o(1)}$, whilst in the other direction there do\nexist such $A$ with $|A| \\ll_C n / \\log^C n$ for any $C$.
"},"Green3.green_3":{"url":"/FormalConjectures/GreensOpenProblems/«3»/#Green3___green_3","anchor":"Green3___green_3","docHtml":"\n Suppose that $A \\subset [0,1]$ is open and has measure greater than $\\frac{1}{3}$. Is there a solution to $xy = z$ with $x, y, z \\in A$?
"},"Green12.green_12":{"url":"/FormalConjectures/GreensOpenProblems/«12»/#Green12___green_12","anchor":"Green12___green_12","docHtml":"\n Let $G$ be an abelian group of size $N$, and suppose that $A \\subset G$ has density $\\alpha$.\nAre there at least $\\alpha^{15} N^{10}$ tuples $(x_1, \\dots, x_5, y_1, \\dots, y_5) \\in G^{10}$\nsuch that $x_i + y_j \\in A$ whenever $j \\in {i, i+1, i+2}$?
\n\n Note: We interpret indices modulo 5.
"},"Green36.SimultaneousDoubleProduct":{"url":"/FormalConjectures/GreensOpenProblems/«36»/#Green36___SimultaneousDoubleProduct","anchor":"Green36___SimultaneousDoubleProduct","docHtml":"\n The simultaneous double product property [CKS05, 4.1].
"},"Green36.Green36Property":{"url":"/FormalConjectures/GreensOpenProblems/«36»/#Green36___Green36Property","anchor":"Green36___Green36Property","docHtml":"\n A variant of the simultaneous double product property, as stated in [Gr24, Problem 36].
"},"Green36.green_36":{"url":"/FormalConjectures/GreensOpenProblems/«36»/#Green36___green_36","anchor":"Green36___green_36","docHtml":"\n Do the following exist, for arbitrarily large $n$? An abelian group $H$ with $|H| = n^{2+o(1)}$,\ntogether with subsets $A_1, ..., A_n, B_1, ..., B_n$ satisfying $|A_i||B_i| \\ge n^{2-o(1)}$ and\n$|A_i + B_i| = |A_i||B_i|$, such that the sets $A_i + B_i$ are disjoint from the sets $A_j + B_k$\n($j \\neq k$)?
\n\n NOTE: according to [CKS05, 4.1], the conditions should be $A_i + B_j$ disjoint from $A_j + B_k$ for\n$i \\neq k$. See green_36.variants.cks05.
\n Variant using the exact simultaneous double product property from [CKS05, 4.1].
"},"Green23.green_23":{"url":"/FormalConjectures/GreensOpenProblems/«23»/#Green23___green_23","anchor":"Green23___green_23","docHtml":"\n Suppose that $\\mathbb{N}$ is finitely coloured. Are there $x,y$ of the same colour such that $x^2 +\ny^2$ is a square?
\n\n Solved in [FrKlMo25].
"},"Green24.max013AffineTranslates":{"url":"/FormalConjectures/GreensOpenProblems/«24»/#Green24___max013AffineTranslates","anchor":"Green24___max013AffineTranslates","docHtml":"\n The maximum number of $\\lbrace 0,1,3 \\rbrace$ affine translates that a set of size $n$ can\ncontain.
"},"Green24.green_24":{"url":"/FormalConjectures/GreensOpenProblems/«24»/#Green24___green_24","anchor":"Green24___green_24","docHtml":"\n If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set\n$\\lbrace 0,1,3 \\rbrace$ that $A$ can contain?
\n\n Conjectured in [Aa19] p.579: $\\left({1}{3} + o(1)\\right) n^2$.
"},"Green24.variants.upper_trivial":{"url":"/FormalConjectures/GreensOpenProblems/«24»/#Green24___variants___upper_trivial","anchor":"Green24___variants___upper_trivial","docHtml":"\n From [Aa19] p.577: the trivial upper bound is $n^2$ (non asymptotic).
"},"Green24.variants.gamma":{"url":"/FormalConjectures/GreensOpenProblems/«24»/#Green24___variants___gamma","anchor":"Green24___variants___gamma","docHtml":"\n The asymptotic constant $\\gamma$ defined in [Aa19] p.579.
"},"Green24.variants.upper_HL":{"url":"/FormalConjectures/GreensOpenProblems/«24»/#Green24___variants___upper_HL","anchor":"Green24___variants___upper_HL","docHtml":"\n Asymptotic upper bound (1.2) in [Aa19]. Named after Hardy and Littlewood [HaL28].
"},"Green24.variants.lower_HL":{"url":"/FormalConjectures/GreensOpenProblems/«24»/#Green24___variants___lower_HL","anchor":"Green24___variants___lower_HL","docHtml":"\n Asymptotic lower bound (1.2) in [Aa19]. Named after Hardy and Littlewood [HaL28].
"},"Green24.variants.conjecture":{"url":"/FormalConjectures/GreensOpenProblems/«24»/#Green24___variants___conjecture","anchor":"Green24___variants___conjecture","docHtml":"\n Conjecture p.579 in [Aa19]: $\\left({1}{3} + o(1)\\right) n^2$.
"},"Green22.HasMonochromaticSumProduct":{"url":"/FormalConjectures/GreensOpenProblems/«22»/#Green22___HasMonochromaticSumProduct","anchor":"Green22___HasMonochromaticSumProduct","docHtml":"\n The monochromatic sum-product property: a colouring $c$ of ${1, \\ldots, N}$ has a pair $(x, y)$\nwith $x, y \\geq 3$ such that $x + y$ and $xy$ are both in ${1, \\ldots, N}$ and receive the same\ncolour.
"},"Green22.N₀":{"url":"/FormalConjectures/GreensOpenProblems/«22»/#Green22___N___","anchor":"Green22___N___","docHtml":"\n $N_0(r)$ is the smallest $N$ such that every $r$-colouring of ${1, \\ldots, N}$ has the\nmonochromatic sum-product property.
"},"Green22.GreenSawhneyBound":{"url":"/FormalConjectures/GreensOpenProblems/«22»/#Green22___GreenSawhneyBound","anchor":"Green22___GreenSawhneyBound","docHtml":"\n The upper bound function from [GrSa25].
"},"Green22.green_22":{"url":"/FormalConjectures/GreensOpenProblems/«22»/#Green22___green_22","anchor":"Green22___green_22","docHtml":"\n If ${1, \\ldots, N}$ is $r$-coloured then, for $N \\geqslant N_0(r)$, there are integers\n$x, y \\geqslant 3$ such that $x + y, xy$ have the same colour.
\n\n Find reasonable bounds for $N_0(r)$. The goal is to improve upon the Green-Sawhney bound.
"},"Green22.green_22.variants.green_sawhney_bound":{"url":"/FormalConjectures/GreensOpenProblems/«22»/#Green22___green_22___variants___green_sawhney_bound","anchor":"Green22___green_22___variants___green_sawhney_bound","docHtml":"\n [GrSa25, Theorem 1.1] found a permissible upper bound.
"},"Green22.green_22.variants.moreira_infinite":{"url":"/FormalConjectures/GreensOpenProblems/«22»/#Green22___green_22___variants___moreira_infinite","anchor":"Green22___green_22___variants___moreira_infinite","docHtml":"\n [Mo17, Corollary 1.5] For any finite coloring of $\\mathbb{N}$ there exist (infinitely many)\n$x, y \\in \\mathbb{N}$ such that ${xy, x + y}$ is monochromatic.
\n\n This guarantees that $N_0(r)$ is well-defined.
\n\n Note: [Mo17] also establishes that $x$ is of the same colour.
"},"Green22.green_22.variants.lower_nine":{"url":"/FormalConjectures/GreensOpenProblems/«22»/#Green22___green_22___variants___lower_nine","anchor":"Green22___green_22___variants___lower_nine","docHtml":"\n Since $x, y \\geq 3$, we must have $xy \\geq 9$, so $N_0(r) \\geq 9$\n(assuming $N_0(r)$ is well-defined, which follows from [Mo17]).
"},"Green29.green_29":{"url":"/FormalConjectures/GreensOpenProblems/«29»/#Green29___green_29","anchor":"Green29___green_29","docHtml":"\n Suppose that $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \\subset A$,\n$|S| \\gg K^{-O(1)} |A|$, with $S^8 \\subset A^4$?
"},"Green29.green_29.variant":{"url":"/FormalConjectures/GreensOpenProblems/«29»/#Green29___green_29___variant","anchor":"Green29___green_29___variant","docHtml":"\n Such a conclusion is known with $|S| \\gg_K |A|$ [Br13 Problem 6.5, CrSi10, Sa10].
"},"Green16.SolutionFree":{"url":"/FormalConjectures/GreensOpenProblems/«16»/#Green16___SolutionFree","anchor":"Green16___SolutionFree","docHtml":"\n A set has no solution to $x + 3y = 2z + 2w$ in distinct elements.
"},"Green16.f":{"url":"/FormalConjectures/GreensOpenProblems/«16»/#Green16___f","anchor":"Green16___f","docHtml":"\n The maximum size of a solution-free subset of $[N]$.
"},"Green16.green_16":{"url":"/FormalConjectures/GreensOpenProblems/«16»/#Green16___green_16","anchor":"Green16___green_16","docHtml":"\n What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?
"},"Green16.green_16_lower_bound":{"url":"/FormalConjectures/GreensOpenProblems/«16»/#Green16___green_16_lower_bound","anchor":"Green16___green_16_lower_bound","docHtml":"\n From [Ruzsa] $f(N) \\gg N^{1/2}$.
"},"Green16.green_16_upper_bound":{"url":"/FormalConjectures/GreensOpenProblems/«16»/#Green16___green_16_upper_bound","anchor":"Green16___green_16_upper_bound","docHtml":"\n From [Schoen and Sisask] $f(N) \\ll N \\cdot e^{-c(\\log N)^{1/7}}$.
"},"Green16.green_16_conjectured_lower_bound":{"url":"/FormalConjectures/GreensOpenProblems/«16»/#Green16___green_16_conjectured_lower_bound","anchor":"Green16___green_16_conjectured_lower_bound","docHtml":"\n $f(N) \\gg N \\cdot e^{-c(\\log N)^{1/7}}$.
"},"Green16.ZhaoSolutionFree":{"url":"/FormalConjectures/GreensOpenProblems/«16»/#Green16___ZhaoSolutionFree","anchor":"Green16___ZhaoSolutionFree","docHtml":"\n A set has no nontrivial solution to $x + 2y + 3z = x' + 2y' + 3z'$.
"},"Green16.g":{"url":"/FormalConjectures/GreensOpenProblems/«16»/#Green16___g","anchor":"Green16___g","docHtml":"\n The maximum size of a Zhao-solution-free subset of $[N]$.
"},"Green16.zhao_question":{"url":"/FormalConjectures/GreensOpenProblems/«16»/#Green16___zhao_question","anchor":"Green16___zhao_question","docHtml":"\n From [Yufei Zhao]: Is there a subset of ${1, \\ldots, N}$ of size\n$N^{1/3 - o(1)}$ with no nontrivial solutions to $x + 2y + 3z = x' + 2y' + 3z'$?
"},"Green32.HasGap":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___HasGap","anchor":"Green32___HasGap","docHtml":"\n A set $A$ has a gap of length $L$ if there exists $x$ such that $x, x+1, \\dots, x+L-1$ are all not\nin $A$.
"},"Green32.hasGap_zero":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___hasGap_zero","anchor":"Green32___hasGap_zero","docHtml":"\n Any set has a gap of length 0 (vacuously true).
"},"Green32.hasGap_empty":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___hasGap_empty","anchor":"Green32___hasGap_empty","docHtml":"\n The empty set has a gap of any length.
"},"Green32.not_hasGap_univ":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___not_hasGap_univ","anchor":"Green32___not_hasGap_univ","docHtml":"\n The full set in $\\mathbb{Z}/p\\mathbb{Z}$ has no gap of positive length.
"},"Green32.hasGap_concrete":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___hasGap_concrete","anchor":"Green32___hasGap_concrete","docHtml":"\n Concrete: ${0}$ in $\\mathbb{Z}/5\\mathbb{Z}$ has a gap of length 4 starting at 1.
"},"Green32.HasLargeGapDilate":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___HasLargeGapDilate","anchor":"Green32___HasLargeGapDilate","docHtml":"\n The generalized problem: for a prime $p$ and a set $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ of size\n$\\lfloor \\omega(p) \\rfloor$, is there a dilate of $A$ containing a gap of length\n$\\lfloor 100p/\\omega(p) \\rfloor$?
"},"Green32.green_32":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___green_32","anchor":"Green32___green_32","docHtml":"\n Let $p$ be a prime and let $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ be a set of size $\\lfloor \\sqrt{p} \\rfloor$.\nIs there a dilate of $A$ containing a gap of length $100\\sqrt{p}$?
"},"Green32.green_32.variants.sh20_general":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___green_32___variants___sh20_general","anchor":"Green32___green_32___variants___sh20_general","docHtml":"\n [Sh20, Theorem 1] implies a gap of at least $\\lfloor 2p/|A| - 2 \\rfloor$.
"},"Green32.green_32.variants.sh20_sqrt":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___green_32___variants___sh20_sqrt","anchor":"Green32___green_32___variants___sh20_sqrt","docHtml":"\n [Sh20] has used the polynomial method to show that this is true with 100 replaced by 2 [Gr24].
\n\n Note: More precisely [Sh20, Theorem 1] implies a gap of at least $\\lfloor 2p/|A| - 2 \\rfloor$.\nFor a set $A$ of size $\\lfloor \\sqrt{p} \\rfloor$, this guarantees a gap of at least\n$\\lfloor 2\\sqrt{p} \\rfloor - 2$.
"},"Green32.green_32.variants.szemeredi_regime":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___green_32___variants___szemeredi_regime","anchor":"Green32___green_32___variants___szemeredi_regime","docHtml":"\n In the regime $\\omega(p) \\sim c p$, this is Szemerédi's theorem [Gr24].
"},"Green32.green_32.variants.dirichlet_regime":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___green_32___variants___dirichlet_regime","anchor":"Green32___green_32___variants___dirichlet_regime","docHtml":"\n In the regime $\\omega(p) \\le c \\log p$, this is basically Dirichlet's lower bound for the size of\nBohr sets [Gr24].
"},"Green32.green_32.variants.log_regime":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___green_32___variants___log_regime","anchor":"Green32___green_32___variants___log_regime","docHtml":"\n Even what happens in the regime $\\omega(p) \\sim 10 \\log p$ is unclear [Gr24].
"},"Green32.HasCosetHole":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___HasCosetHole","anchor":"Green32___HasCosetHole","docHtml":"\n A set $A$ has a coset hole of size $L$ if there exists a subspace $W$ and a vector $v$ such that\nthe affine space $v + W$ has size at least $L$ and is disjoint from $A$.
"},"Green32.hasCosetHole_empty":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___hasCosetHole_empty","anchor":"Green32___hasCosetHole_empty","docHtml":"\n The empty set in $\\mathbb{F}_2^n$ has a coset hole (using the trivial subspace).
"},"Green32.green_32.variants.finite_field":{"url":"/FormalConjectures/GreensOpenProblems/«32»/#Green32___green_32___variants___finite_field","anchor":"Green32___green_32___variants___finite_field","docHtml":"\n Tom Sanders' finite field variant [Gr24].\nIf $N = 2^n$ and $A$ is a subset of size $\\lfloor \\sqrt{N} \\rfloor$, then $A^c$ contains a coset of\nsize at least $100\\sqrt{N}$ for sufficiently large $n$.
"},"Green54.gaussianMeasureInf":{"url":"/FormalConjectures/GreensOpenProblems/«54»/#Green54___gaussianMeasureInf","anchor":"Green54___gaussianMeasureInf","docHtml":"\n The infinite-dimensional Gaussian measure γ∞ on ℝ^ℕ,\ndefined as the countable product of standard Gaussian measures.
"},"Green54.green_54":{"url":"/FormalConjectures/GreensOpenProblems/«54»/#Green54___green_54","anchor":"Green54___green_54","docHtml":"\n Let $K \\subset \\mathbb{R}^n$ be a balanced compact set (that is, $\\lambda K \\subseteq K$ whenever\n$|\\lambda| \\leq 1$) and suppose that the normalised Gaussian measure $\\gamma_n(K) \\geq 0.99$.\nDoes $10K$ contain a compact convex set $C$ with $\\gamma_n(C) \\geq 0.01$?
"},"Green54.green_54_known_case":{"url":"/FormalConjectures/GreensOpenProblems/«54»/#Green54___green_54_known_case","anchor":"Green54___green_54_known_case","docHtml":"\n The same statement is known to be false for 2K instead of 10K.
"},"Green25.Property25":{"url":"/FormalConjectures/GreensOpenProblems/«25»/#Green25___Property25","anchor":"Green25___Property25","docHtml":"\n For which values of $k$ is the following true: whenever we partition $[N] = A_1 \\cup \\dots \\cup A_k$,\n$\\left|\\bigcup^k_{i=1} (A_i \\hat{+} A_i)\\right| \\geq \\frac{1}{10} N$?
"},"Green25.bestLower":{"url":"/FormalConjectures/GreensOpenProblems/«25»/#Green25___bestLower","anchor":"Green25___bestLower","docHtml":"\n The best-known lower bound [ESS89].
"},"Green25.bestUpper":{"url":"/FormalConjectures/GreensOpenProblems/«25»/#Green25___bestUpper","anchor":"Green25___bestUpper","docHtml":"\n The best-known upper bound [ESS89].
"},"Green25.green_25":{"url":"/FormalConjectures/GreensOpenProblems/«25»/#Green25___green_25","anchor":"Green25___green_25","docHtml":"\n For which values of $k$ is the following true: whenever we partition $[N] = A_1 \\cup \\dots \\cup A_k$,\n$\\left|\\bigcup^k_{i=1} (A_i \\hat{+} A_i)\\right| \\geq \\frac{1}{10} N$?
"},"Green25.green_25.upper":{"url":"/FormalConjectures/GreensOpenProblems/«25»/#Green25___green_25___upper","anchor":"Green25___green_25___upper","docHtml":"\n We conjecture that the best-known upper bound can be lowered.
"},"Green25.green_25.lower":{"url":"/FormalConjectures/GreensOpenProblems/«25»/#Green25___green_25___lower","anchor":"Green25___green_25___lower","docHtml":"\n We conjecture that the best-known lower bound can be raised.
"},"Green25.green_25.variants.lower_ess89":{"url":"/FormalConjectures/GreensOpenProblems/«25»/#Green25___green_25___variants___lower_ess89","anchor":"Green25___green_25___variants___lower_ess89","docHtml":"\n For $k \\ll \\log \\log N$, this is true [Gr24].
\n\n NOTE: [ESS89] derive a precise constant for which this is true, but $\\ll$ would imply that it works\nfor any constant. We thus prefer a little-o statement.
"},"Green25.green_25.variants.upper_ess89":{"url":"/FormalConjectures/GreensOpenProblems/«25»/#Green25___green_25___variants___upper_ess89","anchor":"Green25___green_25___variants___upper_ess89","docHtml":"\n For $k \\gg N / \\log N$, it need not be true [Gr24].
\n\n We formalize by enforcing a $N / \\log N$ growth rate on $k(N)$. This avoids cases like $k(N) = N$\nor $N-1$, which produce trivial counter examples with singletons and thus empty restricted sumsets.
"},"Green25.green_25.variants.upper_ess89_trivial":{"url":"/FormalConjectures/GreensOpenProblems/«25»/#Green25___green_25___variants___upper_ess89_trivial","anchor":"Green25___green_25___variants___upper_ess89_trivial","docHtml":"\n For $k \\gg N / \\log N$, it need not be true [Gr24].
\n\n In this version, cases like $k(N) = N$ or $N-1$ can produce trivial counter examples.
"},"Green28.IsUniformOnSupport":{"url":"/FormalConjectures/GreensOpenProblems/«28»/#Green28___IsUniformOnSupport","anchor":"Green28___IsUniformOnSupport","docHtml":"\n True if a PMF on $\\mathbb{Z}$ is uniformly distributed on its support.
"},"Green28.indepSum":{"url":"/FormalConjectures/GreensOpenProblems/«28»/#Green28___indepSum","anchor":"Green28___indepSum","docHtml":"\n The discrete convolution of two PMFs on $\\mathbb{Z}$, representing the distribution of the sum of\ntwo independent random variables.
"},"Green28.green_28":{"url":"/FormalConjectures/GreensOpenProblems/«28»/#Green28___green_28","anchor":"Green28___green_28","docHtml":"\n Suppose that $X, Y$ are two finitely-supported independent random variables taking integer values,\nand such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly\ndistributed on their ranges?
"},"Green2.maxRestrictedSumAvoidingSubsetSize":{"url":"/FormalConjectures/GreensOpenProblems/«2»/#Green2___maxRestrictedSumAvoidingSubsetSize","anchor":"Green2___maxRestrictedSumAvoidingSubsetSize","docHtml":"\n We define the construction from [Sa21, p1] as\n$M(A) := \\max {|S| : S \\subseteq A \\text{ and } (S \\hat{+} S) \\cap A = \\varnothing }$.
"},"Green2.green_2":{"url":"/FormalConjectures/GreensOpenProblems/«2»/#Green2___green_2","anchor":"Green2___green_2","docHtml":"\n Let $A \\subset \\mathbf{Z}$ be a set of $n$ integers. Is there a set $S \\subset A$ of size\n$(\\log n)^{100}$ such that the restricted sumset$S \\hat{+} S$ is disjoint from $A$?
"},"Green2.green_2_lower_bound_sanders":{"url":"/FormalConjectures/GreensOpenProblems/«2»/#Green2___green_2_lower_bound_sanders","anchor":"Green2___green_2_lower_bound_sanders","docHtml":"\n From [Sa21] it is known that there is always such an S with $|S| \\gt (\\log |A|)^{1+c}$.
"},"Green2.green_2_upper_bound_erdos":{"url":"/FormalConjectures/GreensOpenProblems/«2»/#Green2___green_2_upper_bound_erdos","anchor":"Green2___green_2_upper_bound_erdos","docHtml":"\n From [Er65] it is known that $M(A) \\le \\frac{1}{3}|A| + O(1)$.
"},"Green2.green_2_upper_bound_choi":{"url":"/FormalConjectures/GreensOpenProblems/«2»/#Green2___green_2_upper_bound_choi","anchor":"Green2___green_2_upper_bound_choi","docHtml":"\n From [Ch71] it is known that $M(A) \\le |A|^{2/5 + o(1)}$.
"},"Green2.green_2_upper_bound_ruzsa":{"url":"/FormalConjectures/GreensOpenProblems/«2»/#Green2___green_2_upper_bound_ruzsa","anchor":"Green2___green_2_upper_bound_ruzsa","docHtml":"\n From [Ru05] the best-known upper bound is $|S| \\lt e^{C \\sqrt{\\log |A|}}$.
"},"Green44.green_44":{"url":"/FormalConjectures/GreensOpenProblems/«44»/#Green44___green_44","anchor":"Green44___green_44","docHtml":"\n Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes\n$2 \\leqslant p_1 < p_2 < \\dots < p_{1000} < N^{9/10}$. Does the remaining set have size at most\n$\\frac{1}{10} N$?
\n\n We interpret \"half the residue classes\" as $\\lfloor p_i / 2 \\rfloor$.
"},"Green44.green_44.variants.less_than_sqrt":{"url":"/FormalConjectures/GreensOpenProblems/«44»/#Green44___green_44___variants___less_than_sqrt","anchor":"Green44___green_44___variants___less_than_sqrt","docHtml":"\n The answer is affirmative if the primes are all less than $N^{1/2}$, by the large sieve. [Gr24]
"},"LamLitt.IsSolutionOfAlgebraicODE":{"url":"/FormalConjectures/LittProblems/«1»/#LamLitt___IsSolutionOfAlgebraicODE","anchor":"LamLitt___IsSolutionOfAlgebraicODE","docHtml":"\n A power series $f$ is a solution of an algebraic ODE defined by the rational function\n$g \\in \\mathbb{Q}(z, y_0, \\dots, y_{n-1})$ if $f^{(n)}(z) = g(z, f(z), f'(z), \\dots, f^{(n-1)}(z))$.\nThe variable indexed by 0 : Fin (n + 1) corresponds to $z$, and the variable indexed by\ni.succ corresponds to $y_i = f^{(i)}(z)$.
\n There exists $N$ such that for all $n$, the $n$-th coefficient of $f$ is in $\\mathbb{Z}[1/N]$.
"},"LamLitt.omegaIntegral":{"url":"/FormalConjectures/LittProblems/«1»/#LamLitt___omegaIntegral","anchor":"LamLitt___omegaIntegral","docHtml":"\n For an integer-valued function $\\omega$ on the set of primes and a sequence $a_n$ of rational\nnumbers, the condition $\\omega$-integrality means that for each prime $p$, the rational numbers\n$a_0, a_1, \\dots, a_{\\omega(p)}$ are in $\\mathbb{Z}_{(p)}$, i.e. their denominators are not\ndivisible by $p$. When $\\omega(p) < 0$ the constraint at $p$ is vacuous.
"},"LamLitt.omegaSuperlinear":{"url":"/FormalConjectures/LittProblems/«1»/#LamLitt___omegaSuperlinear","anchor":"LamLitt___omegaSuperlinear","docHtml":"\n The growth condition on $\\omega$: the ratio $\\omega(p) / p$ tends to infinity as the prime $p$\ntends to infinity, i.e. $\\lim_{p \\to \\infty} \\omega(p) / p = \\infty$. Here the source filter is\nFilter.atTop on the primes, obtained by pulling back Filter.atTop on $\\mathbb{N}$ along the\ncoercion Nat.Primes → ℕ.
\n Eisenstein's theorem (1852): an algebraic power series over $\\mathbb{Q}[z]$ has bounded\ndenominators, i.e., there exists $N$ such that all coefficients lie in $\\mathbb{Z}[1/N]$.
"},"_private.0.LamLitt.den_coprime_of_mem_adjoinInvNat":{"url":"/FormalConjectures/LittProblems/«1»/#_private___0___LamLitt___den_coprime_of_mem_adjoinInvNat","anchor":"_private___0___LamLitt___den_coprime_of_mem_adjoinInvNat","docHtml":"\n Every element of $\\mathbb{Z}[1/N]$ has denominator coprime to any prime $p$ not in the\nprime factor set of $N$ (vacuously, the case $N = 0$ gives $\\mathbb{Z}[1/N] = \\mathbb{Z}$).
"},"LamLitt.lam_litt.variants.integrality_implies_omega_integrality":{"url":"/FormalConjectures/LittProblems/«1»/#LamLitt___lam_litt___variants___integrality_implies_omega_integrality","anchor":"LamLitt___lam_litt___variants___integrality_implies_omega_integrality","docHtml":"\n Textbook implication: integrality (2) trivially implies ω(p)-integrality (3).
"},"LamLitt.lam_litt.variants.omega_integrality_implies_algebraicity":{"url":"/FormalConjectures/LittProblems/«1»/#LamLitt___lam_litt___variants___omega_integrality_implies_algebraicity","anchor":"LamLitt___lam_litt___variants___omega_integrality_implies_algebraicity","docHtml":"\n implies 2): if the coefficients of $f$ satisfy the $\\omega$-integrality condition for some superlinear $\\omega$,\nthen there exists $N$ such that for all $n$, the $n$-th coefficient of $f$ is in $\\mathbb{Z}[1/N]$.
\n\n implies 1): if the coefficients of $f$ are in $\\mathbb{Z}[1/N]$ for some $N$, then $f$ is algebraic over $\\mathbb{Q}[z]$.\nAlso the version of conjecture of Litt's problem 1 on his website.
\n\nBMO#1
\n\n Let $(a_n)
\n $$(a_{n+1}, b_{n+1}) = \\begin{cases}\n(a_n-b_n, 4b_n+2) & \\text{if }a_n \\ge b_n \\\n(2a_n+1, b_n-a_n) & \\text{if }a_n < b_n\n\\end{cases}$$
\n\n for all positive integers $n$. Does there exist a positive integer $i$ such that $a_i = b_i$?
\n\n The first 10 values of $(a_n, b_n)$ are $(1, 2), (3, 1), (2, 6), (5, 4), (1, 18), (3, 17),\n(7, 14), (15, 7), (8, 30), (17, 22)$.
\n\nBMO#1 is equivalent to asking whether the 6-state Turing machine\n1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE halts or not.
\n There is presently no consensus on whether the machine halts or not, hence the problem is formulated\nusing answer(sorry) ↔.
\n The machine was discovered by bbchallenge.org contributor Jason Yuen on\nJune 25th 2024.
"},"BeaverMathOlympiad.beaver_math_olympiad_problem_2_antihydra":{"url":"/FormalConjectures/Other/BeaverMathOlympiad/#BeaverMathOlympiad___beaver_math_olympiad_problem_2_antihydra","anchor":"BeaverMathOlympiad___beaver_math_olympiad_problem_2_antihydra","docHtml":"\nBMO#2
\n\n Antihydra is a sequence starting at 8, and iterating the function\n$$H(n) = \\left\\lfloor \\frac {3n}2 \\right\\rfloor.$$\nThe conjecture states that the cumulative number of odd values in this sequence\nis never more than twice the cumulative number of even values. It is a relatively new open problem\nwith, so it might be solvable, although seems quite hard because of its Collatz-like flavor.\nThe underlying Collatz-like map has been studied independently in the past,\nsee doi:10.1017/S0017089508004655 (Corollary 4).
\n\n It is equivalent to non-termination of the 1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA 6-state Turing machine (from all-0 tape). Note that the conjecture\nthat the machine does not halt is based on a probabilistic argument.
\n This machine and its mathematical reformulations were found by bbchallenge.org\ncontributors mxdys and Rachel Hunter on June 28th 2024.
"},"BeaverMathOlympiad.beaver_math_olympiad_problem_2_antihydra.variants.set":{"url":"/FormalConjectures/Other/BeaverMathOlympiad/#BeaverMathOlympiad___beaver_math_olympiad_problem_2_antihydra___variants___set","anchor":"BeaverMathOlympiad___beaver_math_olympiad_problem_2_antihydra___variants___set","docHtml":"\nBMO#2 formulation variant
\n\n Alternative statement of beaver_math_olympiad_problem_2_antihydra\nusing set size comparison instead of a recurrent sequence b.
"},"BeaverMathOlympiad.beaver_math_olympiad_problem_3":{"url":"/FormalConjectures/Other/BeaverMathOlympiad/#BeaverMathOlympiad___beaver_math_olympiad_problem_3","anchor":"BeaverMathOlympiad___beaver_math_olympiad_problem_3","docHtml":"\n [BMO#3][https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#3.
\n Let $v_2(n)$ be the largest integer $k$ such that $2^k$ divides $n$.\nLet $(a_n)_{n \\ge 0}$ be a sequence such that
\n\n $$a_n = \\begin{cases}\n2 & \\text{if } n=0 \\\na_{n-1}+2^{v_2(a_{n-1})+2}-1 & \\text{if } n \\ge 1\n\\end{cases}$$
\n\n for all non-negative integers $n$. Is there an integer $n$ such that $a_n=4^k$ for\nsome positive integer $k$?
\n\n [BMO#3][https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#3.1RB0RB3LA4LA2RA_2LB3RA---3RA4RB (from all-0 tape).
\n The machine was found and informally proven not to halt by bbchallenge.org\ncontributor Daniel Yuan on June 18th 2024; see Discord discussion.
"},"BeaverMathOlympiad.beaver_math_olympiad_problem_4":{"url":"/FormalConjectures/Other/BeaverMathOlympiad/#BeaverMathOlympiad___beaver_math_olympiad_problem_4","anchor":"BeaverMathOlympiad___beaver_math_olympiad_problem_4","docHtml":"\nBMO#4
\n\n Bonnie the beaver was bored, so she tried to construct a sequence of integers ${a_n}
\n If $a_n \\equiv 0\\text{ (mod 3)}$, then $a_{n+1}=\\frac{a_n}{3}+2^n+1$.
\n\n If $a_n \\equiv 2\\text{ (mod 3)}$, then $a_{n+1}=\\frac{a_n-2}{3}+2^n-1$.
\n\n With these two rules alone, Bonnie calculates the first few terms in the sequence: $2, 0, 3, 6, 11,\n18, 39, 78, 155, 306, \\dots$. At this point, Bonnie plans to continue writing terms until a term\nbecomes $1\\text{ (mod 3)}$. If Bonnie sticks to her plan, will she ever finish?
\n\nBMO#4\nis equivalent to the non-termination of 2-state 5-symbol Turing machine\n1RB3RB---1LB0LA_2LA4RA3LA4RB1LB (from all-0 tape).
\n The machine was informally proven not to halt bbchallenge.org\ncontributor Daniel Yuan on July 19th 2024; see sketched proof and Discord discussion.
"},"BeaverMathOlympiad.beaver_math_olympiad_problem_5":{"url":"/FormalConjectures/Other/BeaverMathOlympiad/#BeaverMathOlympiad___beaver_math_olympiad_problem_5","anchor":"BeaverMathOlympiad___beaver_math_olympiad_problem_5","docHtml":"\nBMO#5
\n\n Let $(a_n)
\n $$(a_{n+1}, b_{n+1}) = \\begin{cases}\n(a_n+1, b_n-f(a_n)) & \\text{if } b_n \\ge f(a_n) \\\n(a_n, 3b_n+a_n+5) & \\text{if } b_n < f(a_n)\n\\end{cases}$$
\n\n where $f(x)=10\\cdot 2^x-1$ for all non-negative integers $x$.
\n\n Does there exist a positive integer $i$ such that $b_i = f(a_i)-1$?
\n\nBMO#5 is equivalent to asking whether the 6-state Turing machine\n1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE halts or not.
\n There is presently no consensus on whether the machine halts or not, hence the problem is formulated\nusing answer(sorry) ↔.
\n The machine was discovered by bbchallenge.org contributor mxdys\non August 7th 2024.
\n\n The correspondence between the machine's halting problem and the below reformulation has been proven\nin Rocq.
"},"BeaverMathOlympiad.beaver_math_olympiad_problem_8":{"url":"/FormalConjectures/Other/BeaverMathOlympiad/#BeaverMathOlympiad___beaver_math_olympiad_problem_8","anchor":"BeaverMathOlympiad___beaver_math_olympiad_problem_8","docHtml":"\nBMO#8
\n\n Let $(a_n)
\n $$(a_{n+1}, b_{n+1}) = \\begin{cases}\n(a_n - \\lfloor b_n/2 \\rfloor - 3, 3 \\lfloor (b_n+1)/2 \\rfloor + 6) & \\text{if } a_n > \\lfloor b_n/2 \\rfloor \\\n(3 a_n + 5, b_n - 2 a_n) & \\text{if } a_n \\le \\lfloor b_n/2 \\rfloor\n\\end{cases}$$
\n\n for all positive integers $n$. Does there exist a positive integer $i$ such that\n$a_i = \\lfloor b_i/2 \\rfloor + 1$?
\n\nBMO#8 is equivalent to asking whether the 6-state Turing machine\n1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA halts or not.
\n There is presently no consensus on whether the machine halts or not, hence the problem is formulated\nusing answer(sorry) ↔.
\n Every convex set in $\\mathbb R^2$ has VC dimension at most 3.
"},"VCDimConvex.exists_infinite_convex_r3_shatters":{"url":"/FormalConjectures/Other/VCDimConvex/#VCDimConvex___exists_infinite_convex_r3_shatters","anchor":"VCDimConvex___exists_infinite_convex_r3_shatters","docHtml":"\n There exists a set in $\\mathbb R^3$ shattering an infinite set.
"},"VCDimConvex.exists_convex_rn_add_two_vc_n_forall_not_hasAddVCNDimAtMost":{"url":"/FormalConjectures/Other/VCDimConvex/#VCDimConvex___exists_convex_rn_add_two_vc_n_forall_not_hasAddVCNDimAtMost","anchor":"VCDimConvex___exists_convex_rn_add_two_vc_n_forall_not_hasAddVCNDimAtMost","docHtml":"\n There exists a set of infinite $\\mathrm{VC}_n$ dimension in $\\mathbb R^{n + 2}$.
"},"VCDimConvex.hasAddVCNDimAtMost_two_one_of_convex_r3":{"url":"/FormalConjectures/Other/VCDimConvex/#VCDimConvex___hasAddVCNDimAtMost_two_one_of_convex_r3","anchor":"VCDimConvex___hasAddVCNDimAtMost_two_one_of_convex_r3","docHtml":"\n Every convex set in $\\mathbb R^3$ has $\\mathrm{VC}_2$ dimension at most 1.
"},"VCDimConvex.exists_hasAddVCNDimAtMost_n_of_convex_rn_add_one":{"url":"/FormalConjectures/Other/VCDimConvex/#VCDimConvex___exists_hasAddVCNDimAtMost_n_of_convex_rn_add_one","anchor":"VCDimConvex___exists_hasAddVCNDimAtMost_n_of_convex_rn_add_one","docHtml":"\n For every $n$ there exists some $d$ such that every convex set in $\\mathbb R^{n + 1}$ has\n$\\mathrm{VC}_n$ dimension at most $d$.
"},"VCDimConvex.hasAddVCNDimAtMost_n_one_of_convex_rn_add_one":{"url":"/FormalConjectures/Other/VCDimConvex/#VCDimConvex___hasAddVCNDimAtMost_n_one_of_convex_rn_add_one","anchor":"VCDimConvex___hasAddVCNDimAtMost_n_one_of_convex_rn_add_one","docHtml":"\n If $n \\ge 2$, every convex set in $\\mathbb R^{n + 1}$ has $\\mathrm{VC}_n$ dimension at most 1.
"},"SchurTruncatedExponential.truncatedExp":{"url":"/FormalConjectures/Other/SchurTruncatedExponential/#SchurTruncatedExponential___truncatedExp","anchor":"SchurTruncatedExponential___truncatedExp","docHtml":"\n The truncated exponential polynomial truncatedExp n is\ngiven by ∑_{j=0}^{n} x^j / j! over ℚ, which is the\nn-th partial sum of the Taylor series for the exponential function e^x.
\nSchur's Theorem (1924):\nLet f_n(x) = ∑_{j=0}^n x^j/j! be the n-th truncated\nexponential polynomial over ℚ. Then for n ≥ 2:
\n If n ≡ 0 (mod 4), the Galois group of f_n is isomorphic to the alternating group A_n
\n If n ≢ 0 (mod 4), the Galois group of f_n is isomorphic to the symmetric group S_n
\n
\n
\n Equation 255 does not imply Equation 677.
"},"EquationalTheories_677_255.Equation677_not_implies_Equation255":{"url":"/FormalConjectures/Other/EquationalTheories_677_255/#EquationalTheories_677_255___Equation677_not_implies_Equation255","anchor":"EquationalTheories_677_255___Equation677_not_implies_Equation255","docHtml":"\n Equation 677 does not imply Equation 255.
"},"EquationalTheories_677_255.Finite.Equation255_not_implies_Equation677":{"url":"/FormalConjectures/Other/EquationalTheories_677_255/#EquationalTheories_677_255___Finite___Equation255_not_implies_Equation677","anchor":"EquationalTheories_677_255___Finite___Equation255_not_implies_Equation677","docHtml":"\n Note that this is a stronger form of Equation255_not_implies_Equation677.
\n The negation of Finite.Equation677_implies_Equation255.
\n Probably this is true. It would be a stronger form of\nEquation677_not_implies_Equation255.
\n Discussion thread here:\nhttps://leanprover.zulipchat.com/#narrow/channel/458659-Equational/topic/FINITE.3A.20677.20-.3E.20255
"},"EquationalTheories_677_255.Finite.Equation677_implies_Equation255":{"url":"/FormalConjectures/Other/EquationalTheories_677_255/#EquationalTheories_677_255___Finite___Equation677_implies_Equation255","anchor":"EquationalTheories_677_255___Finite___Equation677_implies_Equation255","docHtml":"\n The negation of Finite.Equation677_not_implies_Equation255.
\n Probably this is false.
"},"SuffixPrefixAvoidance.wordSuffix":{"url":"/FormalConjectures/Other/SuffixPrefixAvoidance/#SuffixPrefixAvoidance___wordSuffix","anchor":"SuffixPrefixAvoidance___wordSuffix","docHtml":"\n The suffix of a word w : Fin n → Fin q of length k + 1 (the last k + 1 characters).
\n The prefix of a word w : Fin n → Fin q of length k + 1 (the first k + 1 characters).
\n Two sets of words $A, B$ over Fin q of length n are suffix-prefix avoiding if no\nnonempty suffix of any word in $A$ equals any prefix of any word in $B$ of the same length.
\n $A$ and $B$ are sets of words of length $n$ over alphabet with $q$ letters.\nTrivially then $|A| \\cdot |B|$ is at most $q^{2n}$.
"},"SuffixPrefixAvoidance.suffix_prefix_avoidance_bound":{"url":"/FormalConjectures/Other/SuffixPrefixAvoidance/#SuffixPrefixAvoidance___suffix_prefix_avoidance_bound","anchor":"SuffixPrefixAvoidance___suffix_prefix_avoidance_bound","docHtml":"\n $A$ and $B$ are sets of words of length $n$ over alphabet with $q \\geq 1$ letters.\nNo suffix of a word in $A$ coincides with a prefix of a word in $B$.\nThen $|A| \\cdot |B|$ is at most $\\frac{q^{2n}}{en}$.
\n\n This problem is from
\n $A$ and $B$ are sets of words of length $n$ over alphabet with $q \\geq 1$ letters.\nNo suffix of a word in $A$ coincides with a prefix of a word in $B$.\nThen $|A| \\cdot |B|$ is at most $\\frac{q^{2n}}{n}$.
"},"OpenQuantumProblem35.Config":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___Config","anchor":"OpenQuantumProblem35___Config","docHtml":"\n A computational-basis configuration of $n$ parties with local dimension $d$.
"},"OpenQuantumProblem35.StateVector":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___StateVector","anchor":"OpenQuantumProblem35___StateVector","docHtml":"\n A state vector in the computational basis, viewed as a finite-dimensional Hilbert space.
"},"OpenQuantumProblem35.mkStateVector":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___mkStateVector","anchor":"OpenQuantumProblem35___mkStateVector","docHtml":"\n Build a state vector from its computational-basis amplitudes.
"},"OpenQuantumProblem35.mkStateVector_apply":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___mkStateVector_apply","anchor":"OpenQuantumProblem35___mkStateVector_apply","docHtml":"\n A state built from amplitudes has those amplitudes as its coordinates.
"},"OpenQuantumProblem35.IsNormalized":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___IsNormalized","anchor":"OpenQuantumProblem35___IsNormalized","docHtml":"\n A state vector is normalized if it has $L^2$ norm $1$.
"},"OpenQuantumProblem35.isNormalized_iff_norm_sq_eq_one":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___isNormalized_iff_norm_sq_eq_one","anchor":"OpenQuantumProblem35___isNormalized_iff_norm_sq_eq_one","docHtml":"\n A state is normalized iff its squared $L^2$ norm is $1$.
"},"OpenQuantumProblem35.permuteConfig":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___permuteConfig","anchor":"OpenQuantumProblem35___permuteConfig","docHtml":"\n Permute the parties of a configuration.
"},"OpenQuantumProblem35.permuteConfig_refl":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___permuteConfig_refl","anchor":"OpenQuantumProblem35___permuteConfig_refl","docHtml":"\n The identity permutation leaves a configuration unchanged.
"},"OpenQuantumProblem35.permuteState":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___permuteState","anchor":"OpenQuantumProblem35___permuteState","docHtml":"\n Permute the parties of a state vector.
"},"OpenQuantumProblem35.permuteState_apply":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___permuteState_apply","anchor":"OpenQuantumProblem35___permuteState_apply","docHtml":"\n Evaluating a permuted state vector reads the amplitude at the permuted configuration.
"},"OpenQuantumProblem35.permuteState_refl":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___permuteState_refl","anchor":"OpenQuantumProblem35___permuteState_refl","docHtml":"\n The identity permutation leaves a state vector unchanged.
"},"OpenQuantumProblem35.combineFirst":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___combineFirst","anchor":"OpenQuantumProblem35___combineFirst","docHtml":"\n Merge a configuration on the first $m$ parties and a configuration on the remaining $n-m$\nparties into a configuration on all $n$ parties.
"},"OpenQuantumProblem35.leftIndex":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___leftIndex","anchor":"OpenQuantumProblem35___leftIndex","docHtml":"\n The embedding of the first $m$ indices into $\\mathrm{Fin}, n$.
"},"OpenQuantumProblem35.rightIndex":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___rightIndex","anchor":"OpenQuantumProblem35___rightIndex","docHtml":"\n The embedding of the last $n-m$ indices into $\\mathrm{Fin}, n$.
"},"OpenQuantumProblem35.combineFirst_leftIndex":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___combineFirst_leftIndex","anchor":"OpenQuantumProblem35___combineFirst_leftIndex","docHtml":"\n Combining and then restricting to the left block recovers the left input.
"},"OpenQuantumProblem35.combineFirst_rightIndex":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___combineFirst_rightIndex","anchor":"OpenQuantumProblem35___combineFirst_rightIndex","docHtml":"\n Combining and then restricting to the right block recovers the right input.
"},"OpenQuantumProblem35.reducedDensityFirst":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___reducedDensityFirst","anchor":"OpenQuantumProblem35___reducedDensityFirst","docHtml":"\n The reduced density matrix obtained by tracing out the last $n-m$ parties.
\n\n The subsystem is always the first $m$ parties; different subsystems are handled by first\npermuting the parties.
"},"OpenQuantumProblem35.maximallyMixed":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___maximallyMixed","anchor":"OpenQuantumProblem35___maximallyMixed","docHtml":"\n The maximally mixed state on $m$ parties.
"},"OpenQuantumProblem35.HasMaximallyMixedFirstReduction":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___HasMaximallyMixedFirstReduction","anchor":"OpenQuantumProblem35___HasMaximallyMixedFirstReduction","docHtml":"\n A state has maximally mixed reduction on the first $m$ parties.
"},"OpenQuantumProblem35.IsAME":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___IsAME","anchor":"OpenQuantumProblem35___IsAME","docHtml":"\n A state $\\psi$ is absolutely maximally entangled.
\n\n Standard AME definitions quantify over all subsets $A \\subseteq \\mathrm{Fin}, n$ with\n$|A| \\le \\lfloor n/2 \\rfloor$ and require that the reduction on $A$ be maximally mixed.\nFor pure states it is enough to check subsets of size exactly $\\lfloor n/2 \\rfloor$; see the\nreferences of Helwig--Cui--Riera--Latorre--Lo (2012) and\nGoyeneche--Alsina--Latorre--Riera--Życzkowski (2015). In this file, a subsystem of that size is\nencoded by first permuting the chosen parties to the front and then tracing out the remaining\nparties.
\n\n We also require $\\psi$ to be normalized explicitly.
"},"OpenQuantumProblem35.ExistsAME":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ExistsAME","anchor":"OpenQuantumProblem35___ExistsAME","docHtml":"\n Existence of an $\\mathrm{AME}(n,d)$ state.
"},"OpenQuantumProblem35.not_existsAME_zero_dim":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___not_existsAME_zero_dim","anchor":"OpenQuantumProblem35___not_existsAME_zero_dim","docHtml":"\n No absolutely maximally entangled state exists in local dimension $0$ once $n \\ge 1$.
"},"OpenQuantumProblem35.card_config":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___card_config","anchor":"OpenQuantumProblem35___card_config","docHtml":"\n The number of computational-basis configurations on $m$ parties of local dimension $d$ is $d^m$.
"},"OpenQuantumProblem35.maximallyMixed_apply":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___maximallyMixed_apply","anchor":"OpenQuantumProblem35___maximallyMixed_apply","docHtml":"\n The matrix entries of the maximally mixed state are diagonal and equal to the inverse subsystem dimension.
"},"OpenQuantumProblem35.uniformCoeff":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___uniformCoeff","anchor":"OpenQuantumProblem35___uniformCoeff","docHtml":"\n The common amplitude of the Bell and GHZ witnesses.
"},"OpenQuantumProblem35.IsConstantConfig":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___IsConstantConfig","anchor":"OpenQuantumProblem35___IsConstantConfig","docHtml":"\n A configuration is constant if all coordinates agree.
"},"OpenQuantumProblem35.constantConfig":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___constantConfig","anchor":"OpenQuantumProblem35___constantConfig","docHtml":"\n The constant configuration with value $a$.
"},"OpenQuantumProblem35.isConstantConfig_constantConfig":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___isConstantConfig_constantConfig","anchor":"OpenQuantumProblem35___isConstantConfig_constantConfig","docHtml":"\n Every constant configuration is constant.
"},"OpenQuantumProblem35.not_isConstantConfig_example":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___not_isConstantConfig_example","anchor":"OpenQuantumProblem35___not_isConstantConfig_example","docHtml":"\n A simple binary two-party configuration with different entries is not constant.
"},"OpenQuantumProblem35.diagonalState":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___diagonalState","anchor":"OpenQuantumProblem35___diagonalState","docHtml":"\n The diagonal $n$-party state: the uniform superposition over constant computational-basis strings.
"},"OpenQuantumProblem35.diagonalState_apply":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___diagonalState_apply","anchor":"OpenQuantumProblem35___diagonalState_apply","docHtml":"\n Evaluating the diagonal state returns the uniform coefficient on constant strings and 0 otherwise.
\n The standard $d$-dimensional Bell state.
"},"OpenQuantumProblem35.ghzState":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ghzState","anchor":"OpenQuantumProblem35___ghzState","docHtml":"\n The standard $d$-dimensional GHZ state on $3$ parties.
"},"OpenQuantumProblem35.ghzState4":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ghzState4","anchor":"OpenQuantumProblem35___ghzState4","docHtml":"\n The standard $d$-dimensional GHZ state on $4$ parties.
"},"OpenQuantumProblem35.constantCompletion":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___constantCompletion","anchor":"OpenQuantumProblem35___constantCompletion","docHtml":"\n The completion function for constant-support states reduced to one party.
"},"OpenQuantumProblem35.constantConfig_injective":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___constantConfig_injective","anchor":"OpenQuantumProblem35___constantConfig_injective","docHtml":"\n On a nonempty index type, different constants give different constant configurations.
"},"OpenQuantumProblem35.isConstantConfig_iff_exists_constantConfig":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___isConstantConfig_iff_exists_constantConfig","anchor":"OpenQuantumProblem35___isConstantConfig_iff_exists_constantConfig","docHtml":"\n A configuration on a nonempty index type is constant iff it is equal to some constant configuration.
"},"OpenQuantumProblem35.uniformCoeff_norm_sq":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___uniformCoeff_norm_sq","anchor":"OpenQuantumProblem35___uniformCoeff_norm_sq","docHtml":"\n The squared norm of the uniform coefficient is the inverse local dimension.
"},"OpenQuantumProblem35.uniformCoeff_mul_star":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___uniformCoeff_mul_star","anchor":"OpenQuantumProblem35___uniformCoeff_mul_star","docHtml":"\n The squared norm of the uniform coefficient is the inverse local dimension.
"},"OpenQuantumProblem35.diagonalState_isNormalized":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___diagonalState_isNormalized","anchor":"OpenQuantumProblem35___diagonalState_isNormalized","docHtml":"\n For $n \\ge 1$ and $d \\ge 1$, the diagonal state is normalized.
"},"OpenQuantumProblem35.isConstantConfig_permute_iff":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___isConstantConfig_permute_iff","anchor":"OpenQuantumProblem35___isConstantConfig_permute_iff","docHtml":"\n Permuting the parties preserves the property of being a constant configuration.
"},"OpenQuantumProblem35.diagonalState_permute":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___diagonalState_permute","anchor":"OpenQuantumProblem35___diagonalState_permute","docHtml":"\n The diagonal state is invariant under permutations of the parties.
"},"OpenQuantumProblem35.constantCompletion_eq_iff":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___constantCompletion_eq_iff","anchor":"OpenQuantumProblem35___constantCompletion_eq_iff","docHtml":"\n A tail configuration equals the constant completion of $x$ iff all of its entries agree with the unique entry of $x$.
"},"OpenQuantumProblem35.eq_leftIndex_zero_or_eq_rightIndex":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___eq_leftIndex_zero_or_eq_rightIndex","anchor":"OpenQuantumProblem35___eq_leftIndex_zero_or_eq_rightIndex","docHtml":"\n Every index in $\\mathrm{Fin}, n$ is either the unique left index or a right index when the left block has size $1$.
"},"OpenQuantumProblem35.constantCompletion_injective":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___constantCompletion_injective","anchor":"OpenQuantumProblem35___constantCompletion_injective","docHtml":"\n The completion map for constant configurations is injective once $n \\ge 2$.
"},"OpenQuantumProblem35.isConstantConfig_combineFirst_one_iff":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___isConstantConfig_combineFirst_one_iff","anchor":"OpenQuantumProblem35___isConstantConfig_combineFirst_one_iff","docHtml":"\n A configuration obtained by combining one entry with a tail is constant iff the tail is the constant completion of that entry.
"},"OpenQuantumProblem35.diagonalState_combineFirst_one":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___diagonalState_combineFirst_one","anchor":"OpenQuantumProblem35___diagonalState_combineFirst_one","docHtml":"\n The diagonal state on a split configuration is nonzero exactly on the graph of the constant completion map.
"},"OpenQuantumProblem35.reducedDensityFirst_of_completion":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___reducedDensityFirst_of_completion","anchor":"OpenQuantumProblem35___reducedDensityFirst_of_completion","docHtml":"\n A uniform superposition over the graph of an injective completion map has reduced density matrix $(c\\overline c) I$ on the first subsystem.
"},"OpenQuantumProblem35.hasMaximallyMixedFirstReduction_of_completion":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___hasMaximallyMixedFirstReduction_of_completion","anchor":"OpenQuantumProblem35___hasMaximallyMixedFirstReduction_of_completion","docHtml":"\n The completion criterion gives a maximally mixed reduced state once the coefficient has the correct squared norm.
"},"OpenQuantumProblem35.diagonalState_hasMaximallyMixedFirstReduction_one":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___diagonalState_hasMaximallyMixedFirstReduction_one","anchor":"OpenQuantumProblem35___diagonalState_hasMaximallyMixedFirstReduction_one","docHtml":"\n The diagonal state has maximally mixed one-party reductions once $n \\ge 2$.
"},"OpenQuantumProblem35.diagonalState_isAME_of_div_two_eq_one":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___diagonalState_isAME_of_div_two_eq_one","anchor":"OpenQuantumProblem35___diagonalState_isAME_of_div_two_eq_one","docHtml":"\n If $\\lfloor n/2 \\rfloor = 1$, then the diagonal state is $\\mathrm{AME}(n,d)$ for every $d \\ge 2$.
"},"OpenQuantumProblem35.bellState_isAME":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___bellState_isAME","anchor":"OpenQuantumProblem35___bellState_isAME","docHtml":"\n The standard Bell state is $\\mathrm{AME}(2,d)$ for every physical local dimension $d \\ge 2$.
"},"OpenQuantumProblem35.ghzState_isAME":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ghzState_isAME","anchor":"OpenQuantumProblem35___ghzState_isAME","docHtml":"\n The standard $3$-party GHZ state is $\\mathrm{AME}(3,d)$ for every physical local dimension $d \\ge 2$.
"},"OpenQuantumProblem35.ame_2_exists":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_2_exists","anchor":"OpenQuantumProblem35___ame_2_exists","docHtml":"\n The Bell state witnesses the existence of $\\mathrm{AME}(2,d)$ for every local dimension $d \\ge 2$.
"},"OpenQuantumProblem35.ame_3_exists":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_3_exists","anchor":"OpenQuantumProblem35___ame_3_exists","docHtml":"\n The $3$-party GHZ state witnesses the existence of $\\mathrm{AME}(3,d)$ for every local dimension $d \\ge 2$.
"},"OpenQuantumProblem35.diagonalState_combineFirst_two_of_ne":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___diagonalState_combineFirst_two_of_ne","anchor":"OpenQuantumProblem35___diagonalState_combineFirst_two_of_ne","docHtml":"\n On $4$ parties, the diagonal state vanishes on any split configuration whose first two entries are different.
"},"OpenQuantumProblem35.ghzState4_not_ame":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ghzState4_not_ame","anchor":"OpenQuantumProblem35___ghzState4_not_ame","docHtml":"\n Sanity check: the standard GHZ family on $4$ parties is not absolutely maximally entangled for any local dimension $d \\ge 2$.
"},"OpenQuantumProblem35.ame_2_2_exists":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_2_2_exists","anchor":"OpenQuantumProblem35___ame_2_2_exists","docHtml":"\n Source-backed benchmark statement: the Bell state witnesses the existence of an $\\mathrm{AME}(2,2)$ state.
"},"OpenQuantumProblem35.ame_3_2_exists":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_3_2_exists","anchor":"OpenQuantumProblem35___ame_3_2_exists","docHtml":"\n Source-backed benchmark statement: the three-qubit GHZ state witnesses the existence of an $\\mathrm{AME}(3,2)$ state.
"},"OpenQuantumProblem35.ame_5_2_exists":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_5_2_exists","anchor":"OpenQuantumProblem35___ame_5_2_exists","docHtml":"\n Source-backed benchmark statement: an $\\mathrm{AME}(5,2)$ state exists. This is one of the four qubit cases $n=2,3,5,6$; see the OQP page and Scott (2004).
"},"OpenQuantumProblem35.ame_6_2_exists":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_6_2_exists","anchor":"OpenQuantumProblem35___ame_6_2_exists","docHtml":"\n Source-backed benchmark statement: an $\\mathrm{AME}(6,2)$ state exists. This is one of the four qubit cases $n=2,3,5,6$; see the OQP page and Scott (2004).
"},"OpenQuantumProblem35.ame_4_2_not_exists":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_4_2_not_exists","anchor":"OpenQuantumProblem35___ame_4_2_not_exists","docHtml":"\n Source-backed benchmark statement: no $\\mathrm{AME}(4,2)$ state exists; see Higuchi--Sudbery (2000) and the OQP page.
"},"OpenQuantumProblem35.ame_7_2_not_exists":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_7_2_not_exists","anchor":"OpenQuantumProblem35___ame_7_2_not_exists","docHtml":"\n Source-backed benchmark statement: no $\\mathrm{AME}(7,2)$ state exists; see Huber--Gühne--Siewert (2017) and the OQP page.
"},"OpenQuantumProblem35.ame_4_3_exists":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_4_3_exists","anchor":"OpenQuantumProblem35___ame_4_3_exists","docHtml":"\n Source-backed benchmark statement: an $\\mathrm{AME}(4,3)$ state exists; see Helwig et al. (2012) and Goyeneche et al. (2015).
"},"OpenQuantumProblem35.ame_4_6_exists":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_4_6_exists","anchor":"OpenQuantumProblem35___ame_4_6_exists","docHtml":"\n Source-backed benchmark statement: an $\\mathrm{AME}(4,6)$ state exists; see Rather et al. (2022).
"},"OpenQuantumProblem35.ame_7_6_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_7_6_open","anchor":"OpenQuantumProblem35___ame_7_6_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(7,6)$ state exist?
"},"OpenQuantumProblem35.ame_7_10_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_7_10_open","anchor":"OpenQuantumProblem35___ame_7_10_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(7,10)$ state exist?
"},"OpenQuantumProblem35.ame_8_4_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_8_4_open","anchor":"OpenQuantumProblem35___ame_8_4_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(8,4)$ state exist?
"},"OpenQuantumProblem35.ame_8_6_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_8_6_open","anchor":"OpenQuantumProblem35___ame_8_6_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(8,6)$ state exist?
"},"OpenQuantumProblem35.ame_8_10_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_8_10_open","anchor":"OpenQuantumProblem35___ame_8_10_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(8,10)$ state exist?
"},"OpenQuantumProblem35.ame_9_6_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_9_6_open","anchor":"OpenQuantumProblem35___ame_9_6_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(9,6)$ state exist?
"},"OpenQuantumProblem35.ame_9_10_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_9_10_open","anchor":"OpenQuantumProblem35___ame_9_10_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(9,10)$ state exist?
"},"OpenQuantumProblem35.ame_10_6_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_10_6_open","anchor":"OpenQuantumProblem35___ame_10_6_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(10,6)$ state exist?
"},"OpenQuantumProblem35.ame_10_10_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_10_10_open","anchor":"OpenQuantumProblem35___ame_10_10_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(10,10)$ state exist?
"},"OpenQuantumProblem35.ame_11_3_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_11_3_open","anchor":"OpenQuantumProblem35___ame_11_3_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(11,3)$ state exist?
"},"OpenQuantumProblem35.ame_11_4_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_11_4_open","anchor":"OpenQuantumProblem35___ame_11_4_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(11,4)$ state exist?
"},"OpenQuantumProblem35.ame_11_5_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_11_5_open","anchor":"OpenQuantumProblem35___ame_11_5_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(11,5)$ state exist?
\n\n The DeepMind prover agent has shown that such a state exists.
"},"OpenQuantumProblem35.ame_11_6_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_11_6_open","anchor":"OpenQuantumProblem35___ame_11_6_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(11,6)$ state exist?
"},"OpenQuantumProblem35.ame_11_10_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_11_10_open","anchor":"OpenQuantumProblem35___ame_11_10_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(11,10)$ state exist?
"},"OpenQuantumProblem35.ame_12_5_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_12_5_open","anchor":"OpenQuantumProblem35___ame_12_5_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(12,5)$ state exist?
"},"OpenQuantumProblem35.ame_12_6_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_12_6_open","anchor":"OpenQuantumProblem35___ame_12_6_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(12,6)$ state exist?
"},"OpenQuantumProblem35.ame_12_10_open":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___ame_12_10_open","anchor":"OpenQuantumProblem35___ame_12_10_open","docHtml":"\n Open benchmark statement: does an $\\mathrm{AME}(12,10)$ state exist?
"},"OpenQuantumProblem35.oqp_35":{"url":"/FormalConjectures/OpenQuantumProblems/«35»/#OpenQuantumProblem35___oqp_35","anchor":"OpenQuantumProblem35___oqp_35","docHtml":"\n Open Quantum Problem 35: classify all pairs $(n,d)$ with $n \\ge 2$ and $d \\ge 2$ for which an $\\mathrm{AME}(n,d)$ state exists.
"},"OpenQuantumProblem23.StateVector":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___StateVector","anchor":"OpenQuantumProblem23___StateVector","docHtml":"\n A state vector in the $d$-dimensional complex Hilbert space $\\mathbb{C}^d$.
"},"OpenQuantumProblem23.mkStateVector":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___mkStateVector","anchor":"OpenQuantumProblem23___mkStateVector","docHtml":"\n Build a state vector from its coordinates in the computational basis.
"},"OpenQuantumProblem23.IsNormalized":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___IsNormalized","anchor":"OpenQuantumProblem23___IsNormalized","docHtml":"\n A state vector is normalized if it has $L^2$ norm $1$.
"},"OpenQuantumProblem23.overlapSq":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___overlapSq","anchor":"OpenQuantumProblem23___overlapSq","docHtml":"\n The squared magnitude of the overlap between two state vectors.
"},"OpenQuantumProblem23.HasConstantOverlapSq":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___HasConstantOverlapSq","anchor":"OpenQuantumProblem23___HasConstantOverlapSq","docHtml":"\n A family has constant pairwise squared overlap $c$ if every two distinct members have squared overlap $c$.
"},"OpenQuantumProblem23.sicOverlapSq":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___sicOverlapSq","anchor":"OpenQuantumProblem23___sicOverlapSq","docHtml":"\n The squared overlap value of a SIC family in dimension $d$.
"},"OpenQuantumProblem23.IsSICFamily":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___IsSICFamily","anchor":"OpenQuantumProblem23___IsSICFamily","docHtml":"\n A SIC family in dimension $d$ consists of $d^2$ normalized vectors in $\\mathbb{C}^d$ with pairwise squared overlap $(d + 1)^{-1}$.
"},"OpenQuantumProblem23.HasSICPOVM":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___HasSICPOVM","anchor":"OpenQuantumProblem23___HasSICPOVM","docHtml":"\n There exists a SIC-POVM in dimension $d$.
"},"OpenQuantumProblem23.hasConstantOverlapSq_singleton":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasConstantOverlapSq_singleton","anchor":"OpenQuantumProblem23___hasConstantOverlapSq_singleton","docHtml":"\n Any singleton family has constant pairwise squared overlap, vacuously.
"},"OpenQuantumProblem23.sicOverlapSq_one":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___sicOverlapSq_one","anchor":"OpenQuantumProblem23___sicOverlapSq_one","docHtml":"\n The SIC overlap value in dimension $1$ is $1/2$.
"},"OpenQuantumProblem23.sicOverlapSq_pos":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___sicOverlapSq_pos","anchor":"OpenQuantumProblem23___sicOverlapSq_pos","docHtml":"\n The SIC overlap value is positive in every dimension.
"},"OpenQuantumProblem23.isSICFamily_singleton_iff":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___isSICFamily_singleton_iff","anchor":"OpenQuantumProblem23___isSICFamily_singleton_iff","docHtml":"\n In dimension $1$, a singleton family is SIC exactly when its vector is normalized.
"},"OpenQuantumProblem23.hasSICPOVM_zero":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_zero","anchor":"OpenQuantumProblem23___hasSICPOVM_zero","docHtml":"\n The empty family witnesses the degenerate dimension-$0$ case.
"},"OpenQuantumProblem23.isSICFamily_one_of_normalized":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___isSICFamily_one_of_normalized","anchor":"OpenQuantumProblem23___isSICFamily_one_of_normalized","docHtml":"\n Any normalized state in dimension $1$ yields a SIC family.
"},"OpenQuantumProblem23.hasSICPOVM_one":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_one","anchor":"OpenQuantumProblem23___hasSICPOVM_one","docHtml":"\n Dimension $1$ admits a SIC-POVM.
"},"OpenQuantumProblem23.ω":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23______","anchor":"OpenQuantumProblem23______","docHtml":"\n The standard algebraic primitive cube root of unity.
"},"OpenQuantumProblem23.tetraA":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___tetraA","anchor":"OpenQuantumProblem23___tetraA","docHtml":"\n The first real amplitude used in the tetrahedral qubit SIC.
"},"OpenQuantumProblem23.tetraB":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___tetraB","anchor":"OpenQuantumProblem23___tetraB","docHtml":"\n The second real amplitude used in the tetrahedral qubit SIC.
"},"OpenQuantumProblem23.hesseS":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hesseS","anchor":"OpenQuantumProblem23___hesseS","docHtml":"\n The common scale used in the Hesse qutrit SIC.
"},"OpenQuantumProblem23.vec2":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___vec2","anchor":"OpenQuantumProblem23___vec2","docHtml":"\n A convenient constructor for qubit state vectors.
"},"OpenQuantumProblem23.vec3":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___vec3","anchor":"OpenQuantumProblem23___vec3","docHtml":"\n A convenient constructor for qutrit state vectors.
"},"OpenQuantumProblem23.qubitSICFamily":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___qubitSICFamily","anchor":"OpenQuantumProblem23___qubitSICFamily","docHtml":"\n The tetrahedral qubit SIC family.
"},"OpenQuantumProblem23.hesseFamily":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hesseFamily","anchor":"OpenQuantumProblem23___hesseFamily","docHtml":"\n The Hesse qutrit SIC family.
"},"OpenQuantumProblem23.bb84Family":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___bb84Family","anchor":"OpenQuantumProblem23___bb84Family","docHtml":"\n The BB84 family of four qubit states.
"},"OpenQuantumProblem23.sicOverlapSq_two":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___sicOverlapSq_two","anchor":"OpenQuantumProblem23___sicOverlapSq_two","docHtml":"\n The SIC overlap value in dimension $2$ is $1/3$.
"},"OpenQuantumProblem23.sicOverlapSq_three":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___sicOverlapSq_three","anchor":"OpenQuantumProblem23___sicOverlapSq_three","docHtml":"\n The SIC overlap value in dimension $3$ is $1/4$.
"},"OpenQuantumProblem23.qubitSICFamily_normalized":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___qubitSICFamily_normalized","anchor":"OpenQuantumProblem23___qubitSICFamily_normalized","docHtml":"\n Every vector in the tetrahedral qubit SIC family is normalized.
"},"OpenQuantumProblem23.qubitSICFamily_pairwise":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___qubitSICFamily_pairwise","anchor":"OpenQuantumProblem23___qubitSICFamily_pairwise","docHtml":"\n The tetrahedral qubit SIC family has the correct constant pairwise overlap.
"},"OpenQuantumProblem23.hasSICPOVM_two":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_two","anchor":"OpenQuantumProblem23___hasSICPOVM_two","docHtml":"\n Dimension $2$ admits a SIC-POVM, witnessed by the tetrahedral qubit SIC.
"},"_private.0.OpenQuantumProblem23.omega_norm":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#_private___0___OpenQuantumProblem23___omega_norm","anchor":"_private___0___OpenQuantumProblem23___omega_norm","docHtml":"\n The unit complex number ω (primitive cube root) has norm 1.
\n Every vector in the Hesse qutrit SIC family is normalized.
"},"OpenQuantumProblem23.hesseFamily_pairwise":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hesseFamily_pairwise","anchor":"OpenQuantumProblem23___hesseFamily_pairwise","docHtml":"\n The Hesse qutrit SIC family has the correct constant pairwise overlap.
"},"OpenQuantumProblem23.hasSICPOVM_three":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_three","anchor":"OpenQuantumProblem23___hasSICPOVM_three","docHtml":"\n Dimension $3$ admits a SIC-POVM, witnessed by the Hesse qutrit SIC.
"},"OpenQuantumProblem23.bb84Family_normalized":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___bb84Family_normalized","anchor":"OpenQuantumProblem23___bb84Family_normalized","docHtml":"\n Every vector in the BB84 family is normalized.
"},"OpenQuantumProblem23.bb84Family_not_isSICFamily":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___bb84Family_not_isSICFamily","anchor":"OpenQuantumProblem23___bb84Family_not_isSICFamily","docHtml":"\n The BB84 family has the right cardinality for a qubit SIC but fails the constant-overlap condition.
"},"OpenQuantumProblem23.hasSICPOVM_56":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_56","anchor":"OpenQuantumProblem23___hasSICPOVM_56","docHtml":"\n Benchmark open subproblem: existence of a SIC-POVM in dimension $56$.
"},"OpenQuantumProblem23.hasSICPOVM_58":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_58","anchor":"OpenQuantumProblem23___hasSICPOVM_58","docHtml":"\n Benchmark open subproblem: existence of a SIC-POVM in dimension $58$.
"},"OpenQuantumProblem23.hasSICPOVM_59":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_59","anchor":"OpenQuantumProblem23___hasSICPOVM_59","docHtml":"\n Benchmark open subproblem: existence of a SIC-POVM in dimension $59$.
"},"OpenQuantumProblem23.hasSICPOVM_60":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_60","anchor":"OpenQuantumProblem23___hasSICPOVM_60","docHtml":"\n Benchmark open subproblem: existence of a SIC-POVM in dimension $60$.
"},"OpenQuantumProblem23.hasSICPOVM_64":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_64","anchor":"OpenQuantumProblem23___hasSICPOVM_64","docHtml":"\n Benchmark open subproblem: existence of a SIC-POVM in dimension $64$.
"},"OpenQuantumProblem23.hasSICPOVM_68":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_68","anchor":"OpenQuantumProblem23___hasSICPOVM_68","docHtml":"\n Benchmark open subproblem: existence of a SIC-POVM in dimension $68$.
"},"OpenQuantumProblem23.hasSICPOVM_69":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_69","anchor":"OpenQuantumProblem23___hasSICPOVM_69","docHtml":"\n Benchmark open subproblem: existence of a SIC-POVM in dimension $69$.
"},"OpenQuantumProblem23.hasSICPOVM_70":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_70","anchor":"OpenQuantumProblem23___hasSICPOVM_70","docHtml":"\n Benchmark open subproblem: existence of a SIC-POVM in dimension $70$.
"},"OpenQuantumProblem23.hasSICPOVM_71":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_71","anchor":"OpenQuantumProblem23___hasSICPOVM_71","docHtml":"\n Benchmark open subproblem: existence of a SIC-POVM in dimension $71$.
"},"OpenQuantumProblem23.hasSICPOVM_72":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_72","anchor":"OpenQuantumProblem23___hasSICPOVM_72","docHtml":"\n Benchmark open subproblem: existence of a SIC-POVM in dimension $72$.
"},"OpenQuantumProblem23.hasSICPOVM_75":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___hasSICPOVM_75","anchor":"OpenQuantumProblem23___hasSICPOVM_75","docHtml":"\n Benchmark open subproblem: existence of a SIC-POVM in dimension $75$.
"},"OpenQuantumProblem23.sicPOVMs":{"url":"/FormalConjectures/OpenQuantumProblems/«23»/#OpenQuantumProblem23___sicPOVMs","anchor":"OpenQuantumProblem23___sicPOVMs","docHtml":"\n Do SIC-POVMs exist in every finite dimension?
"},"OpenQuantumProblem13.UMat":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___UMat","anchor":"OpenQuantumProblem13___UMat","docHtml":"\n A unitary matrix representing an orthonormal basis of $\\mathbb{C}^d$ via its columns.
"},"OpenQuantumProblem13.relativeUnitary":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___relativeUnitary","anchor":"OpenQuantumProblem13___relativeUnitary","docHtml":"\n The relative unitary between two bases.
"},"OpenQuantumProblem13.IsUnbiased":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___IsUnbiased","anchor":"OpenQuantumProblem13___IsUnbiased","docHtml":"\n Two unitary matrices represent mutually unbiased bases if every entry of the relative\nunitary has squared norm $1 / d$.
"},"OpenQuantumProblem13.IsUnbiased.symm":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___IsUnbiased___symm","anchor":"OpenQuantumProblem13___IsUnbiased___symm","docHtml":"\n Mutual unbiasedness is symmetric.
"},"OpenQuantumProblem13.IsMUBFamily":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___IsMUBFamily","anchor":"OpenQuantumProblem13___IsMUBFamily","docHtml":"\n A family of unitary matrices is a family of mutually unbiased bases if every two distinct\nmembers are unbiased.
"},"OpenQuantumProblem13.HasMUBs":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___HasMUBs","anchor":"OpenQuantumProblem13___HasMUBs","docHtml":"\n There exist $k$ mutually unbiased bases in $\\mathbb{C}^d$.
"},"OpenQuantumProblem13.HasCompleteMUBs":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___HasCompleteMUBs","anchor":"OpenQuantumProblem13___HasCompleteMUBs","docHtml":"\n There exists a complete set of $d + 1$ mutually unbiased bases in $\\mathbb{C}^d$.
"},"OpenQuantumProblem13.IsMaxMUBCount":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___IsMaxMUBCount","anchor":"OpenQuantumProblem13___IsMaxMUBCount","docHtml":"\n $k$ is the maximal size of a family of mutually unbiased bases in dimension $d$.
"},"OpenQuantumProblem13.hasMUBs_zero":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___hasMUBs_zero","anchor":"OpenQuantumProblem13___hasMUBs_zero","docHtml":"\n Every dimension admits the empty family of mutually unbiased bases.
"},"OpenQuantumProblem13.hasMUBs_one":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___hasMUBs_one","anchor":"OpenQuantumProblem13___hasMUBs_one","docHtml":"\n Every dimension admits a family of one mutually unbiased basis.
"},"OpenQuantumProblem13.Qubit.ω":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit______","anchor":"OpenQuantumProblem13___Qubit______","docHtml":"\n A convenient phase with squared norm $1/2$. Using $\\omega = (1+i)/2$ avoids square roots.
"},"OpenQuantumProblem13.Qubit.phaseMatrix":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___phaseMatrix","anchor":"OpenQuantumProblem13___Qubit___phaseMatrix","docHtml":"\n The raw phase-parametrized Hadamard matrix. The cases $\\zeta = 1$ and $\\zeta = i$\ngive the $X$ and $Y$ bases.
"},"OpenQuantumProblem13.Qubit.omega_norm_sq":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___omega_norm_sq","anchor":"OpenQuantumProblem13___Qubit___omega_norm_sq","docHtml":"\n The squared norm of ω is $1/2$.
\n The product $\\overline{\\omega},\\omega$ is $1/2$.
"},"OpenQuantumProblem13.Qubit.star_smul_mul_smul":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___star_smul_mul_smul","anchor":"OpenQuantumProblem13___Qubit___star_smul_mul_smul","docHtml":"\n Taking the star of a scalar multiple on the left and multiplying by another scalar multiple\ncollects the scalar factor as $\\overline{a} a$.
"},"OpenQuantumProblem13.Qubit.star_phaseMatrix_mul_phaseMatrix":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___star_phaseMatrix_mul_phaseMatrix","anchor":"OpenQuantumProblem13___Qubit___star_phaseMatrix_mul_phaseMatrix","docHtml":"\n The relative product of two phase matrices has the expected $2 \\times 2$ form.
"},"OpenQuantumProblem13.Qubit.star_phaseMatrix_mul_self_of_unit_phase":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___star_phaseMatrix_mul_self_of_unit_phase","anchor":"OpenQuantumProblem13___Qubit___star_phaseMatrix_mul_self_of_unit_phase","docHtml":"\n If $\\zeta$ has unit modulus, then the phase matrix is orthogonal up to the scalar factor $2$.
"},"OpenQuantumProblem13.Qubit.scaled_phaseMatrix_mem_unitary":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___scaled_phaseMatrix_mem_unitary","anchor":"OpenQuantumProblem13___Qubit___scaled_phaseMatrix_mem_unitary","docHtml":"\n Scaling a phase matrix by $\\omega$ produces a unitary matrix whenever the phase has unit modulus.
"},"OpenQuantumProblem13.Qubit.star_phaseBasis_mul_phaseBasis":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___star_phaseBasis_mul_phaseBasis","anchor":"OpenQuantumProblem13___Qubit___star_phaseBasis_mul_phaseBasis","docHtml":"\n The relative product of two scaled phase matrices is obtained by scaling the corresponding\nrelative product of phase matrices.
"},"OpenQuantumProblem13.Qubit.phaseU":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___phaseU","anchor":"OpenQuantumProblem13___Qubit___phaseU","docHtml":"\n The bundled qubit basis associated to a unit-modulus phase $\\zeta$.
"},"OpenQuantumProblem13.Qubit.phase_norm_sq_eq_one":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___phase_norm_sq_eq_one","anchor":"OpenQuantumProblem13___Qubit___phase_norm_sq_eq_one","docHtml":"\n A complex number with $\\overline{\\zeta},\\zeta = 1$ has squared norm $1$.
"},"OpenQuantumProblem13.Qubit.omega_mul_phase_norm_sq":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___omega_mul_phase_norm_sq","anchor":"OpenQuantumProblem13___Qubit___omega_mul_phase_norm_sq","docHtml":"\n Multiplying $\\omega$ by a unit-modulus phase preserves the squared norm $1/2$.
"},"OpenQuantumProblem13.Qubit.ZU":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___ZU","anchor":"OpenQuantumProblem13___Qubit___ZU","docHtml":"\n The standard qubit basis.
"},"OpenQuantumProblem13.Qubit.XU":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___XU","anchor":"OpenQuantumProblem13___Qubit___XU","docHtml":"\n The qubit $X$ basis as a bundled unitary matrix.
"},"OpenQuantumProblem13.Qubit.YU":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___YU","anchor":"OpenQuantumProblem13___Qubit___YU","docHtml":"\n The qubit $Y$ basis as a bundled unitary matrix.
"},"OpenQuantumProblem13.Qubit.isUnbiased_Z_phaseU":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___isUnbiased_Z_phaseU","anchor":"OpenQuantumProblem13___Qubit___isUnbiased_Z_phaseU","docHtml":"\n The standard basis is mutually unbiased with any phase basis of unit-modulus phase.
"},"OpenQuantumProblem13.Qubit.relative_phaseU_phaseU_of_mul_eq_I":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___relative_phaseU_phaseU_of_mul_eq_I","anchor":"OpenQuantumProblem13___Qubit___relative_phaseU_phaseU_of_mul_eq_I","docHtml":"\n If $\\overline{\\zeta},\\eta = i$, then the relative unitary between the corresponding phase\nbases is the qubit mutually unbiased overlap matrix.
"},"OpenQuantumProblem13.Qubit.isUnbiased_phaseU_phaseU_of_mul_eq_I":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___isUnbiased_phaseU_phaseU_of_mul_eq_I","anchor":"OpenQuantumProblem13___Qubit___isUnbiased_phaseU_phaseU_of_mul_eq_I","docHtml":"\n If $\\overline{\\zeta},\\eta = i$, then the corresponding phase bases are mutually unbiased.
"},"OpenQuantumProblem13.Qubit.qubitFamily":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___qubitFamily","anchor":"OpenQuantumProblem13___Qubit___qubitFamily","docHtml":"\n The three standard qubit mutually unbiased bases: $Z$, $X$, and $Y$.
"},"OpenQuantumProblem13.Qubit.qubitFamily_isMUB":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___qubitFamily_isMUB","anchor":"OpenQuantumProblem13___Qubit___qubitFamily_isMUB","docHtml":"\n The standard qubit family is a family of mutually unbiased bases.
"},"OpenQuantumProblem13.Qubit.qubit_hasThreeMUBs":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___qubit_hasThreeMUBs","anchor":"OpenQuantumProblem13___Qubit___qubit_hasThreeMUBs","docHtml":"\n There exist three mutually unbiased bases in dimension $2$.
"},"OpenQuantumProblem13.Qubit.u0":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___u0","anchor":"OpenQuantumProblem13___Qubit___u0","docHtml":"\n The first entry of the first column of a qubit unitary basis matrix.
"},"OpenQuantumProblem13.Qubit.u1":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___u1","anchor":"OpenQuantumProblem13___Qubit___u1","docHtml":"\n The second entry of the first column of a qubit unitary basis matrix.
"},"OpenQuantumProblem13.Qubit.BlochVec":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___BlochVec","anchor":"OpenQuantumProblem13___Qubit___BlochVec","docHtml":"\n The real Bloch-vector space for qubits.
"},"OpenQuantumProblem13.Qubit.bloch":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___bloch","anchor":"OpenQuantumProblem13___Qubit___bloch","docHtml":"\n The Bloch vector associated to the first column of a qubit basis matrix.
"},"OpenQuantumProblem13.Qubit.relativeUnitary_apply_zero_zero":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___relativeUnitary_apply_zero_zero","anchor":"OpenQuantumProblem13___Qubit___relativeUnitary_apply_zero_zero","docHtml":"\n The $(0,0)$ entry of the relative unitary is the overlap of the first columns.
"},"OpenQuantumProblem13.Qubit.firstCol_normSq":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___firstCol_normSq","anchor":"OpenQuantumProblem13___Qubit___firstCol_normSq","docHtml":"\n The first column of a unitary matrix has squared norm $1$.
"},"OpenQuantumProblem13.Qubit.re_mul_conj":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___re_mul_conj","anchor":"OpenQuantumProblem13___Qubit___re_mul_conj","docHtml":"\n The real part of $z \\overline{w}$ is the Euclidean dot product of the coordinate pairs of\nz and w.
\n The Bloch inner product is determined by the $(0,0)$ entry of the relative unitary.
"},"OpenQuantumProblem13.Qubit.relativeUnitary_self":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___relativeUnitary_self","anchor":"OpenQuantumProblem13___Qubit___relativeUnitary_self","docHtml":"\n The relative unitary of a basis with itself is the identity matrix.
"},"OpenQuantumProblem13.Qubit.bloch_inner_self":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___bloch_inner_self","anchor":"OpenQuantumProblem13___Qubit___bloch_inner_self","docHtml":"\n Every qubit Bloch vector has squared Euclidean norm $1$.
"},"OpenQuantumProblem13.Qubit.bloch_ne_zero":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___bloch_ne_zero","anchor":"OpenQuantumProblem13___Qubit___bloch_ne_zero","docHtml":"\n A qubit Bloch vector is never the zero vector.
"},"OpenQuantumProblem13.Qubit.bloch_inner_eq_zero_of_isUnbiased":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___bloch_inner_eq_zero_of_isUnbiased","anchor":"OpenQuantumProblem13___Qubit___bloch_inner_eq_zero_of_isUnbiased","docHtml":"\n Mutually unbiased qubit bases have orthogonal Bloch vectors.
"},"OpenQuantumProblem13.Qubit.qubit_upper_bound":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___qubit_upper_bound","anchor":"OpenQuantumProblem13___Qubit___qubit_upper_bound","docHtml":"\n No family of mutually unbiased bases in dimension $2$ has size greater than $3$.
"},"OpenQuantumProblem13.Qubit.qubit_maximal":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___Qubit___qubit_maximal","anchor":"OpenQuantumProblem13___Qubit___qubit_maximal","docHtml":"\n The maximum number of mutually unbiased bases in dimension $2$ is $3$.
"},"OpenQuantumProblem13.mutuallyUnbiasedBases_dim2":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___mutuallyUnbiasedBases_dim2","anchor":"OpenQuantumProblem13___mutuallyUnbiasedBases_dim2","docHtml":"\n In dimension $2$, the maximum number of mutually unbiased orthonormal bases is $3$.
"},"OpenQuantumProblem13.mutuallyUnbiasedBases_dim6_bounds":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___mutuallyUnbiasedBases_dim6_bounds","anchor":"OpenQuantumProblem13___mutuallyUnbiasedBases_dim6_bounds","docHtml":"\n Known general bounds in dimension $6$: the maximal number of mutually unbiased bases\nsatisfies $3 \\le \\mu(6) \\le 7$.
"},"OpenQuantumProblem13.mutuallyUnbiasedBases_dim6":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___mutuallyUnbiasedBases_dim6","anchor":"OpenQuantumProblem13___mutuallyUnbiasedBases_dim6","docHtml":"\n Special case in dimension $6$: determine the maximal number of mutually unbiased\northonormal bases in $\\mathbb{C}^6$.
"},"OpenQuantumProblem13.mutuallyUnbiasedBases_dim10":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___mutuallyUnbiasedBases_dim10","anchor":"OpenQuantumProblem13___mutuallyUnbiasedBases_dim10","docHtml":"\n Special case in dimension $10$ (not a prime power): determine the maximal number of\nmutually unbiased orthonormal bases in $\\mathbb{C}^{10}$.
"},"OpenQuantumProblem13.mutuallyUnbiasedBases_dim12":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___mutuallyUnbiasedBases_dim12","anchor":"OpenQuantumProblem13___mutuallyUnbiasedBases_dim12","docHtml":"\n Special case in dimension $12$ (not a prime power): determine the maximal number of\nmutually unbiased orthonormal bases in $\\mathbb{C}^{12}$.
"},"OpenQuantumProblem13.mutuallyUnbiasedBases_dim14":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___mutuallyUnbiasedBases_dim14","anchor":"OpenQuantumProblem13___mutuallyUnbiasedBases_dim14","docHtml":"\n Special case in dimension $14$ (not a prime power): determine the maximal number of\nmutually unbiased orthonormal bases in $\\mathbb{C}^{14}$.
"},"OpenQuantumProblem13.mutuallyUnbiasedBases_dim15":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___mutuallyUnbiasedBases_dim15","anchor":"OpenQuantumProblem13___mutuallyUnbiasedBases_dim15","docHtml":"\n Special case in dimension $15$ (not a prime power): determine the maximal number of\nmutually unbiased orthonormal bases in $\\mathbb{C}^{15}$.
"},"OpenQuantumProblem13.mutuallyUnbiasedBases":{"url":"/FormalConjectures/OpenQuantumProblems/«13»/#OpenQuantumProblem13___mutuallyUnbiasedBases","anchor":"OpenQuantumProblem13___mutuallyUnbiasedBases","docHtml":"\n Open Quantum Problem 13: determine the maximal number of mutually unbiased orthonormal\nbases in $\\mathbb{C}^d$ for $d \\ge 2$.
"},"WrittenOnTheWallII.GraphConjecture13.conjecture13":{"url":"/FormalConjectures/WrittenOnTheWallII/GraphConjecture13/#WrittenOnTheWallII___GraphConjecture13___conjecture13","anchor":"WrittenOnTheWallII___GraphConjecture13___conjecture13","docHtml":"\n WOWII Conjecture 13
\n\n For a simple connected graph G, the size b(G) of a largest induced bipartite subgraph\nsatisfies b(G) ≥ diam(G) + max_{v ∈ V} l(v) - 1, where diam(G) is the diameter\nof G and l(v) = indepNeighborsCard G v is the independence number of the\nneighbourhood of v.
\n WOWII Conjecture 327
\n\n Let G be a simple connected graph. If 3 · γ(G) = γ_i(G), then G is well\ntotally dominated, where γ(G) is the domination number of G and γ_i(G) is\nthe independent domination number of G.
\nProof Sketch:\nThe conjecture states that if $3\\gamma(G) = i(G)$ for a connected graph $G$, then $G$ is well totally dominated.\nHowever, this conjecture is FALSE.
\n\nCounterexample:\nConsider a graph $G$ with 12 vertices: $u, v, a_0, a_1, a_2, a_3, a_4, b_0, b_1, b_2, b_3, b_4$.\nThe edges are:
\n\n $(u, v)$
\n\n $(u, a_i)$ for all $i \\in {0, 1, 2, 3, 4}$
\n\n $(v, b_i)$ for all $i \\in {0, 1, 2, 3, 4}$
\n\n $(a_0, b_3), (a_1, b_3), (a_2, b_0), (a_3, b_0), (a_4, b_3), (a_4, b_4)$
\n\n Properties of $G$:
\n\nConnected: Yes, path exists between any two vertices through $u$ and $v$.
\n\nDomination Number $\\gamma(G)$: The set ${u, v}$ dominates all vertices. Since there is no universal vertex, $\\gamma(G) = 2$.
\n\nIndependent Domination Number $i(G)$: The minimum independent dominating set has size 6 (e.g., ${u, b_0, b_1, b_2, b_3, b_4}$). Thus $i(G) = 6$.
\n\nCondition: $3 \\gamma(G) = 3 \\times 2 = 6 = i(G)$. The condition holds.
\n\nWell Totally Dominated: A graph is well totally dominated if all minimal total dominating sets have the same size.
\n\n ${u, v}$ is a minimal total dominating set of size 2.
\n\n ${v, b_0, b_3}$ is a minimal total dominating set of size 3.\nSince $2 \\neq 3$, $G$ is NOT well totally dominated.
\n\n The counterexample has been found by Moritz Firsching and Goran Žužić using an\nexperimental pipeline.
"},"WrittenOnTheWallII.Test.HouseGraph":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___HouseGraph","anchor":"WrittenOnTheWallII___Test___HouseGraph","docHtml":"\n House Graph: Square 0-1-2-3-0 with roof 4 connected to 2,3.
"},"WrittenOnTheWallII.Test.K4":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4","anchor":"WrittenOnTheWallII___Test___K4","docHtml":"\n K4: Complete graph on 4 vertices.
"},"WrittenOnTheWallII.Test.PetersenGraph":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___PetersenGraph","anchor":"WrittenOnTheWallII___Test___PetersenGraph","docHtml":"\n Petersen Graph on 10 vertices.
"},"WrittenOnTheWallII.Test.C6":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6","anchor":"WrittenOnTheWallII___Test___C6","docHtml":"\n C6: Cycle graph on 6 vertices.
"},"WrittenOnTheWallII.Test.Star5":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5","anchor":"WrittenOnTheWallII___Test___Star5","docHtml":"\n Star5: Star graph with center 0 and 5 leaves.
"},"WrittenOnTheWallII.Test.house_indep":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_indep","anchor":"WrittenOnTheWallII___Test___house_indep"},"WrittenOnTheWallII.Test.house_dom":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_dom","anchor":"WrittenOnTheWallII___Test___house_dom"},"WrittenOnTheWallII.Test.house_avg_dist":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_avg_dist","anchor":"WrittenOnTheWallII___Test___house_avg_dist"},"WrittenOnTheWallII.Test.house_diameter":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_diameter","anchor":"WrittenOnTheWallII___Test___house_diameter"},"WrittenOnTheWallII.Test.house_radius":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_radius","anchor":"WrittenOnTheWallII___Test___house_radius"},"WrittenOnTheWallII.Test.house_girth":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_girth","anchor":"WrittenOnTheWallII___Test___house_girth"},"WrittenOnTheWallII.Test.house_order":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_order","anchor":"WrittenOnTheWallII___Test___house_order"},"WrittenOnTheWallII.Test.house_size":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_size","anchor":"WrittenOnTheWallII___Test___house_size"},"WrittenOnTheWallII.Test.house_szeged":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_szeged","anchor":"WrittenOnTheWallII___Test___house_szeged"},"WrittenOnTheWallII.Test.house_wiener":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_wiener","anchor":"WrittenOnTheWallII___Test___house_wiener"},"WrittenOnTheWallII.Test.house_min_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_min_deg","anchor":"WrittenOnTheWallII___Test___house_min_deg"},"WrittenOnTheWallII.Test.house_max_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_max_deg","anchor":"WrittenOnTheWallII___Test___house_max_deg"},"WrittenOnTheWallII.Test.house_avg_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_avg_deg","anchor":"WrittenOnTheWallII___Test___house_avg_deg"},"WrittenOnTheWallII.Test.house_matching":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_matching","anchor":"WrittenOnTheWallII___Test___house_matching"},"WrittenOnTheWallII.Test.house_residue":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_residue","anchor":"WrittenOnTheWallII___Test___house_residue"},"WrittenOnTheWallII.Test.house_annihilation":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_annihilation","anchor":"WrittenOnTheWallII___Test___house_annihilation"},"WrittenOnTheWallII.Test.house_cvetkovic":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___house_cvetkovic","anchor":"WrittenOnTheWallII___Test___house_cvetkovic"},"WrittenOnTheWallII.Test.K4_indep":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_indep","anchor":"WrittenOnTheWallII___Test___K4_indep"},"WrittenOnTheWallII.Test.K4_dom":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_dom","anchor":"WrittenOnTheWallII___Test___K4_dom"},"WrittenOnTheWallII.Test.K4_avg_dist":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_avg_dist","anchor":"WrittenOnTheWallII___Test___K4_avg_dist"},"WrittenOnTheWallII.Test.K4_diameter":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_diameter","anchor":"WrittenOnTheWallII___Test___K4_diameter"},"WrittenOnTheWallII.Test.K4_radius":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_radius","anchor":"WrittenOnTheWallII___Test___K4_radius"},"WrittenOnTheWallII.Test.K4_girth":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_girth","anchor":"WrittenOnTheWallII___Test___K4_girth"},"WrittenOnTheWallII.Test.K4_order":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_order","anchor":"WrittenOnTheWallII___Test___K4_order"},"WrittenOnTheWallII.Test.K4_size":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_size","anchor":"WrittenOnTheWallII___Test___K4_size"},"WrittenOnTheWallII.Test.K4_szeged":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_szeged","anchor":"WrittenOnTheWallII___Test___K4_szeged"},"WrittenOnTheWallII.Test.K4_wiener":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_wiener","anchor":"WrittenOnTheWallII___Test___K4_wiener"},"WrittenOnTheWallII.Test.K4_min_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_min_deg","anchor":"WrittenOnTheWallII___Test___K4_min_deg"},"WrittenOnTheWallII.Test.K4_max_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_max_deg","anchor":"WrittenOnTheWallII___Test___K4_max_deg"},"WrittenOnTheWallII.Test.K4_avg_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_avg_deg","anchor":"WrittenOnTheWallII___Test___K4_avg_deg"},"WrittenOnTheWallII.Test.K4_matching":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_matching","anchor":"WrittenOnTheWallII___Test___K4_matching"},"WrittenOnTheWallII.Test.K4_residue":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_residue","anchor":"WrittenOnTheWallII___Test___K4_residue"},"WrittenOnTheWallII.Test.K4_annihilation":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_annihilation","anchor":"WrittenOnTheWallII___Test___K4_annihilation"},"WrittenOnTheWallII.Test.K4_cvetkovic":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___K4_cvetkovic","anchor":"WrittenOnTheWallII___Test___K4_cvetkovic"},"WrittenOnTheWallII.Test.petersen_indep":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_indep","anchor":"WrittenOnTheWallII___Test___petersen_indep"},"WrittenOnTheWallII.Test.petersen_dom":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_dom","anchor":"WrittenOnTheWallII___Test___petersen_dom"},"WrittenOnTheWallII.Test.petersen_avg_dist":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_avg_dist","anchor":"WrittenOnTheWallII___Test___petersen_avg_dist"},"WrittenOnTheWallII.Test.petersen_diameter":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_diameter","anchor":"WrittenOnTheWallII___Test___petersen_diameter"},"WrittenOnTheWallII.Test.petersen_radius":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_radius","anchor":"WrittenOnTheWallII___Test___petersen_radius"},"WrittenOnTheWallII.Test.petersen_girth":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_girth","anchor":"WrittenOnTheWallII___Test___petersen_girth"},"WrittenOnTheWallII.Test.petersen_order":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_order","anchor":"WrittenOnTheWallII___Test___petersen_order"},"WrittenOnTheWallII.Test.petersen_size":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_size","anchor":"WrittenOnTheWallII___Test___petersen_size"},"WrittenOnTheWallII.Test.petersen_szeged":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_szeged","anchor":"WrittenOnTheWallII___Test___petersen_szeged"},"WrittenOnTheWallII.Test.petersen_wiener":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_wiener","anchor":"WrittenOnTheWallII___Test___petersen_wiener"},"WrittenOnTheWallII.Test.petersen_min_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_min_deg","anchor":"WrittenOnTheWallII___Test___petersen_min_deg"},"WrittenOnTheWallII.Test.petersen_max_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_max_deg","anchor":"WrittenOnTheWallII___Test___petersen_max_deg"},"WrittenOnTheWallII.Test.petersen_avg_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_avg_deg","anchor":"WrittenOnTheWallII___Test___petersen_avg_deg"},"WrittenOnTheWallII.Test.petersen_matching":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_matching","anchor":"WrittenOnTheWallII___Test___petersen_matching"},"WrittenOnTheWallII.Test.petersen_residue":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_residue","anchor":"WrittenOnTheWallII___Test___petersen_residue"},"WrittenOnTheWallII.Test.petersen_annihilation":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_annihilation","anchor":"WrittenOnTheWallII___Test___petersen_annihilation"},"WrittenOnTheWallII.Test.petersen_cvetkovic":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___petersen_cvetkovic","anchor":"WrittenOnTheWallII___Test___petersen_cvetkovic"},"WrittenOnTheWallII.Test.C6_indep":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_indep","anchor":"WrittenOnTheWallII___Test___C6_indep"},"WrittenOnTheWallII.Test.C6_dom":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_dom","anchor":"WrittenOnTheWallII___Test___C6_dom"},"WrittenOnTheWallII.Test.C6_avg_dist":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_avg_dist","anchor":"WrittenOnTheWallII___Test___C6_avg_dist"},"WrittenOnTheWallII.Test.C6_diameter":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_diameter","anchor":"WrittenOnTheWallII___Test___C6_diameter"},"WrittenOnTheWallII.Test.C6_radius":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_radius","anchor":"WrittenOnTheWallII___Test___C6_radius"},"WrittenOnTheWallII.Test.C6_girth":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_girth","anchor":"WrittenOnTheWallII___Test___C6_girth"},"WrittenOnTheWallII.Test.C6_order":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_order","anchor":"WrittenOnTheWallII___Test___C6_order"},"WrittenOnTheWallII.Test.C6_size":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_size","anchor":"WrittenOnTheWallII___Test___C6_size"},"WrittenOnTheWallII.Test.C6_szeged":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_szeged","anchor":"WrittenOnTheWallII___Test___C6_szeged"},"WrittenOnTheWallII.Test.C6_wiener":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_wiener","anchor":"WrittenOnTheWallII___Test___C6_wiener"},"WrittenOnTheWallII.Test.C6_min_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_min_deg","anchor":"WrittenOnTheWallII___Test___C6_min_deg"},"WrittenOnTheWallII.Test.C6_max_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_max_deg","anchor":"WrittenOnTheWallII___Test___C6_max_deg"},"WrittenOnTheWallII.Test.C6_avg_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_avg_deg","anchor":"WrittenOnTheWallII___Test___C6_avg_deg"},"WrittenOnTheWallII.Test.C6_matching":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_matching","anchor":"WrittenOnTheWallII___Test___C6_matching"},"WrittenOnTheWallII.Test.C6_residue":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_residue","anchor":"WrittenOnTheWallII___Test___C6_residue"},"WrittenOnTheWallII.Test.C6_annihilation":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_annihilation","anchor":"WrittenOnTheWallII___Test___C6_annihilation"},"WrittenOnTheWallII.Test.C6_cvetkovic":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___C6_cvetkovic","anchor":"WrittenOnTheWallII___Test___C6_cvetkovic"},"WrittenOnTheWallII.Test.Star5_indep":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_indep","anchor":"WrittenOnTheWallII___Test___Star5_indep"},"WrittenOnTheWallII.Test.Star5_dom":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_dom","anchor":"WrittenOnTheWallII___Test___Star5_dom"},"WrittenOnTheWallII.Test.Star5_avg_dist":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_avg_dist","anchor":"WrittenOnTheWallII___Test___Star5_avg_dist"},"WrittenOnTheWallII.Test.Star5_diameter":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_diameter","anchor":"WrittenOnTheWallII___Test___Star5_diameter"},"WrittenOnTheWallII.Test.Star5_radius":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_radius","anchor":"WrittenOnTheWallII___Test___Star5_radius"},"WrittenOnTheWallII.Test.Star5_girth":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_girth","anchor":"WrittenOnTheWallII___Test___Star5_girth"},"WrittenOnTheWallII.Test.Star5_order":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_order","anchor":"WrittenOnTheWallII___Test___Star5_order"},"WrittenOnTheWallII.Test.Star5_size":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_size","anchor":"WrittenOnTheWallII___Test___Star5_size"},"WrittenOnTheWallII.Test.Star5_szeged":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_szeged","anchor":"WrittenOnTheWallII___Test___Star5_szeged"},"WrittenOnTheWallII.Test.Star5_wiener":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_wiener","anchor":"WrittenOnTheWallII___Test___Star5_wiener"},"WrittenOnTheWallII.Test.Star5_min_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_min_deg","anchor":"WrittenOnTheWallII___Test___Star5_min_deg"},"WrittenOnTheWallII.Test.Star5_max_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_max_deg","anchor":"WrittenOnTheWallII___Test___Star5_max_deg"},"WrittenOnTheWallII.Test.Star5_avg_deg":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_avg_deg","anchor":"WrittenOnTheWallII___Test___Star5_avg_deg"},"WrittenOnTheWallII.Test.Star5_matching":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_matching","anchor":"WrittenOnTheWallII___Test___Star5_matching"},"WrittenOnTheWallII.Test.Star5_residue":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_residue","anchor":"WrittenOnTheWallII___Test___Star5_residue"},"WrittenOnTheWallII.Test.Star5_annihilation":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_annihilation","anchor":"WrittenOnTheWallII___Test___Star5_annihilation"},"WrittenOnTheWallII.Test.Star5_cvetkovic":{"url":"/FormalConjectures/WrittenOnTheWallII/Test/#WrittenOnTheWallII___Test___Star5_cvetkovic","anchor":"WrittenOnTheWallII___Test___Star5_cvetkovic"},"WrittenOnTheWallII.GraphConjecture19.conjecture19":{"url":"/FormalConjectures/WrittenOnTheWallII/GraphConjecture19/#WrittenOnTheWallII___GraphConjecture19___conjecture19","anchor":"WrittenOnTheWallII___GraphConjecture19___conjecture19","docHtml":"\n WOWII Conjecture 19
\n\n If G is connected then the size b(G) of a largest induced bipartite subgraph\nsatisfies\nb(G) ≥ FLOOR((∑ ecc(v))/(|V|) + sSup (range (l G))), where ecc(v) denotes\neccentricity and l(G) is the independence number of neighbourhoods.
\n WOWII Conjecture 194
\n\n For a simple connected graph G, if α(G) ≤ 1 + l_avg(G), then G has a Hamiltonian path.\nHere α(G) = G.indepNum is the independence number, and\nl_avg(G) = averageIndepNeighbors G is the average over all vertices of the independence number\nof the neighbourhood.\nA Hamiltonian path is a walk visiting every vertex exactly once.
\n WOWII Conjecture 23
\n\n For a simple connected graph G, b(G) ≥ ⌊α(G) + dist_avg(M, V) / 2⌋, where b(G) is\nthe size of a largest induced bipartite subgraph, α(G) is the independence number,\nand M is the set of maximum-degree vertices, and dist_avg(M, V) is the average\ndistance from all vertices to M.
\n This conjecture is false; there is a counterexample with b(G) = 19, α(G) = 15,\nand dist_avg(M, V) = 10.
\n WOWII Conjecture 34
\n\n For a simple connected graph G, path(G) ≥ ⌈dist_avg(C, V) + dist_avg(M, V)⌉,\nwhere path(G) is the floor of the average distance of G, C is the set of center\nvertices (those with minimum eccentricity), M is the set of maximum-degree vertices,\nand dist_avg(S, V) is the average distance from all vertices to the set S.
\n WOWII Conjecture 32
\n\n For a simple connected graph $G$,\n$\\operatorname{path}(G) \\ge \\operatorname{dist}_{\\operatorname{avg}}(A) + 0.5 \\cdot \\operatorname{ecc}_{\\operatorname{avg}}(M)$,\nwhere $\\operatorname{path}(G)$ is the floor of the average distance of $G$, $A$ is the\nset of minimum-degree vertices, $M$ is the set of maximum-degree vertices,\n$\\operatorname{dist}_{\\operatorname{avg}}(A)$ is the average distance from all vertices\nto $A$, and $\\operatorname{ecc}_{\\operatorname{avg}}(M)$ is the average eccentricity\nof the vertices in $M$.
\n\n The conjecture is false, the authors present a counterexample: \"The path on 5 vertices\nis a counterexample, path = 5, distavg(A) = 4 and the average of eccentricity of maximum\ndegree vertices is 8/3.\"
"},"WrittenOnTheWallII.GraphConjecture4.conjecture4":{"url":"/FormalConjectures/WrittenOnTheWallII/GraphConjecture4/#WrittenOnTheWallII___GraphConjecture4___conjecture4","anchor":"WrittenOnTheWallII___GraphConjecture4___conjecture4","docHtml":"\n WOWII Conjecture 4
\n\n If G is a connected graph then the maximum number of leaves over all spanning\ntrees satisfies Ls(G) ≥ NG(G) - 1 where NG(G) is the minimal neighbourhood\nsize of a non-edge of G.
\n WOWII Conjecture 20
\n\n For a simple connected graph G, b(G) ≥ n(G) / ⌊deg_avg(G)⌋, where b(G) is\nthe size of a largest induced bipartite subgraph, n(G) is the number of vertices,\nand deg_avg(G) = (∑ v, deg(v)) / n(G) is the average degree.
\n WOWII Conjecture 5
\n\n For a simple connected graph G, Ls(G) is bounded below by the maximal size\nof a sphere of radius radius(G) around the centres of G.
\n WOWII Conjecture 2
\n\n For a simple connected graph G,\nLs(G) ≥ 2 · (l(G) - 1) where l(G) is the average independence number of\nthe neighbourhoods of the vertices of G.
\n WOWII Conjecture 315
\n\n Let G be a simple connected graph and let P denote the set of pendant vertices\n(vertices of degree 1). If α(G) = |P|, then G is well totally dominated.
\n WOWII Conjecture 3
\n\n For a connected simple graph G, the number of leaves in a maximum spanning\ntree satisfies Ls(G) ≥ G.indepDominationNumber * MaxTemp(G), where G.indepDominationNumber is the independent\ndomination number and MaxTemp(G) is max_v deg(v)/(n(G) - deg(v)).
\n WOWII Conjecture 322
\n\n Let G be a simple connected graph on n ≥ 5 vertices. If the maximum over all\nvertices v of l(v) — the independence number of the neighborhood N(v) of v\n— is at most 1, then G is well totally dominated.
\n Here l(v) = α(G[N(v)]) is the independence number of the subgraph induced by the\nopen neighborhood of v.
\n WOWII Conjecture 198a
\n\n For a simple connected graph G, if b(G) ≤ 2 + ecc_avg(G), then G has a Hamiltonian path.\nHere b(G) is the number of vertices in a largest induced bipartite subgraph, and\necc_avg(G) is the average eccentricity of G.\nA Hamiltonian path is a walk visiting every vertex exactly once.
\n WOWII Conjecture 1
\n\n For a simple connected graph G the maximum number of leaves of a spanning\ntree satisfies Ls(G) ≥ n(G) + 1 - 2·m(G) where n(G) counts vertices and\nm(G) is the size of a maximum matching.
\n WOWII Conjecture 33
\n\n For a simple connected graph G, path(G) ≥ ⌈2 · dist_avg(M, V)⌉, where path(G)\nis the floor of the average distance of G, M is the set of maximum-degree vertices,\nand dist_avg(M, V) is the average distance from all vertices to M.
\n WOWII Conjecture 200
\n\n For a simple connected graph G, if tree(G) = ⌈1 + l_avg(G)⌉, then G has a Hamiltonian path.\nHere tree(G) is the number of vertices of a largest induced tree subgraph, and\nl_avg(G) = averageIndepNeighbors G is the average over all vertices of the independence number\nof the neighbourhood.\nA Hamiltonian path is a walk visiting every vertex exactly once.
\n WOWII Conjecture 6
\n\n For a connected graph G we have\nLs(G) ≥ 1 + n(G) - m(G) - a(G) where a(G) is defined via independent sets.
\n WOWII Conjecture 40
\n\n For a nontrivial connected graph G the size f(G) of a largest induced forest\nsatisfies f(G) ≥ ceil((p(G) + b(G) + 1)/2) where p(G) is the path cover\nnumber and b(G) is the largest induced bipartite subgraph size.
\n WOWII Conjecture 316
\n\n Let G be a simple connected graph and let P denote the set of pendant vertices\n(vertices of degree 1). If |P| ≥ deg_avg(Gᶜ), then G is well totally dominated,\nwhere deg_avg(Gᶜ) is the average degree of the complement of G.
\n WOWII Conjecture 16
\n\n For a simple connected graph G, the size b(G) of a largest induced bipartite subgraph\nsatisfies b(G) ≥ 2 * (rad(G) - 1) + max_{v ∈ V} l(v), where rad(G) is the radius\nof G (the minimum eccentricity) and l(v) = indepNeighborsCard G v is the independence\nnumber of the neighbourhood of v.
\n WOWII Conjecture 141
\n\n For a simple connected graph G,\ntree(G) ≥ ⌊girth(G) / 2⌋ - 1 + max_v l(v)\nwhere tree(G) is the number of vertices of a largest induced tree subgraph,\ngirth(G) is the length of the shortest cycle (0 if acyclic), and\nl(v) = indepNeighbors G v is the independence number of the neighbourhood of v.
\n WOWII Conjecture 17
\n\n For a simple connected graph G, the size b(G) of a largest induced bipartite subgraph\nsatisfies b(G) ≥ α(G) + ⌈diam(G) / 3⌉, where α(G) is the independence number of G\nand diam(G) is the diameter of G.
\n WOWII Conjecture 58
\n\n For a connected graph G, the size f(G) of a largest induced forest satisfies\nf(G) ≥ ceil( b(G) / average l(v) ) where b(G) is the largest induced\nbipartite subgraph and l(v) is the independence number of G.neighborSet v.
\n Vertices of $K_N$.
"},"MonochromaticQuantumGraph.EdgeN":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___EdgeN","anchor":"MonochromaticQuantumGraph___EdgeN","docHtml":"\n Edge label for $K_N$ with endpoint indices in Fin D.
\n We u < v (so undirected edges are represented once),\nand our enumeration always pairs the first vertex in an ordered list with a later vertex.
\n Edge label for $K_N$ with endpoint indices in Fin D.
\n We u < v (so undirected edges are represented once),\nand our enumeration always pairs the first vertex in an ordered list with a later vertex.
\n Edge label for $K_N$ with endpoint indices in Fin D.
\n We u < v (so undirected edges are represented once),\nand our enumeration always pairs the first vertex in an ordered list with a later vertex.
\n Edge label for $K_N$ with endpoint indices in Fin D.
\n We u < v (so undirected edges are represented once),\nand our enumeration always pairs the first vertex in an ordered list with a later vertex.
\n Edge label for $K_N$ with endpoint indices in Fin D.
\n We u < v (so undirected edges are represented once),\nand our enumeration always pairs the first vertex in an ordered list with a later vertex.
\n Weights on edges.
"},"MonochromaticQuantumGraph.mkEdge":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___mkEdge","anchor":"MonochromaticQuantumGraph___mkEdge","docHtml":"\n Helper: build an EdgeN from endpoints and endpoint indices.
\n Ordered vertex list $[0, 1, \\ldots, N-1]$.
"},"MonochromaticQuantumGraph.allEqualList":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___allEqualList","anchor":"MonochromaticQuantumGraph___allEqualList","docHtml":"\n Chain-equality along a list of vertices.
"},"MonochromaticQuantumGraph.allEqual":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___allEqual","anchor":"MonochromaticQuantumGraph___allEqual","docHtml":"\n All indices equal on Fin N (using the canonical ordered vertex list).
\n Auxiliary perfect-matching sum on a vertex list, using a fuel parameter n for termination.
\n When called as pmSumListAux W ι L.length L, this computes the weighted sum over all perfect\nmatchings on the vertices in L. The recursion pairs the head vertex with each later vertex and\nrecurses on the remaining vertices.
\n For lists of odd length, there are no perfect matchings and the value is 0.
\n Perfect-matching sum on a list: run pmSumListAux with fuel = L.length.
\n The perfect-matching sum for $K_N$: use the canonical ordered vertex list vertices N.
\n The monochromatic quantum graph equation system for $K_N$.
\n\n For every index assignment $\\iota : V_N \\to \\mathrm{Fin}, D$, the perfect-matching sum equals $1$\nif $\\iota$ is constant (monochromatic inherited vertex colouring), and equals $0$ otherwise.
"},"MonochromaticQuantumGraph.instDecidableEqSystemN":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___instDecidableEqSystemN","anchor":"MonochromaticQuantumGraph___instDecidableEqSystemN","docHtml":"\n Instance: EqSystemN N D W is decidable when α has decidable equality.
\n Sanity check over ℕ using native_decide.
\n Sanity check over ℕ using native_decide.
\n Sanity check over ℕ using native_decide.
\n Bogdanov: for $N = 4$ and all $D \\geq 4$, no solution exists over $\\mathbb{R}_{\\geq 0}$.
"},"MonochromaticQuantumGraph.eqSystem_no_solution_nnreal_even_ge6_ge3":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem_no_solution_nnreal_even_ge6_ge3","anchor":"MonochromaticQuantumGraph___eqSystem_no_solution_nnreal_even_ge6_ge3","docHtml":"\n Bogdanov: for all even $N \\geq 6$ and $D \\geq 3$, no solution exists over $\\mathbb{R}_{\\geq 0}$.
"},"MonochromaticQuantumGraph.eqSystem_no_solution_even_ge4_d_eq_n_explicit":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem_no_solution_even_ge4_d_eq_n_explicit","anchor":"MonochromaticQuantumGraph___eqSystem_no_solution_even_ge4_d_eq_n_explicit","docHtml":"\n For all even $N \\geq 4$ and $D = N$, does there exist no solution to the monochromatic quantum\ngraph equation system over $\\mathbb{C}$?
\n\n The DeepMind prover agent has found a formal proof for this statement.
"},"MonochromaticQuantumGraph.eqSystem4_no_solution_d4":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem4_no_solution_d4","anchor":"MonochromaticQuantumGraph___eqSystem4_no_solution_d4","docHtml":"\n For $N = 4$ and $D = 4$, does there exist no solution to the monochromatic quantum\ngraph equation system over $\\mathbb{C}$?
\n\n This is the $D = N$ case, proved using eqSystem_no_solution_even_ge4_d_eq_n_explicit.
\n For $N = 4$ and all $D \\geq 4$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
\n\n The DeepMind prover agent has found a formal proof of this statement.
"},"MonochromaticQuantumGraph.eqSystem6_no_solution_d3":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem6_no_solution_d3","anchor":"MonochromaticQuantumGraph___eqSystem6_no_solution_d3","docHtml":"\n For $N = 6$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem6_no_solution_d4":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem6_no_solution_d4","anchor":"MonochromaticQuantumGraph___eqSystem6_no_solution_d4","docHtml":"\n For $N = 6$ and $D = 4$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem6_no_solution_d5":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem6_no_solution_d5","anchor":"MonochromaticQuantumGraph___eqSystem6_no_solution_d5","docHtml":"\n For $N = 6$ and $D = 5$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem6_no_solution_d6":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem6_no_solution_d6","anchor":"MonochromaticQuantumGraph___eqSystem6_no_solution_d6","docHtml":"\n For $N = 6$ and $D = 6$, does there exist no solution to the monochromatic quantum\ngraph equation system over $\\mathbb{C}$?
\n\n This follows from eqSystem_no_solution_even_ge4_d_eq_n_explicit.
\n For $N = 6$ and all $D \\geq 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem8_no_solution_d3":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem8_no_solution_d3","anchor":"MonochromaticQuantumGraph___eqSystem8_no_solution_d3","docHtml":"\n For $N = 8$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem8_no_solution_d10":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem8_no_solution_d10","anchor":"MonochromaticQuantumGraph___eqSystem8_no_solution_d10","docHtml":"\n For $N = 8$ and $D = 10$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
\n\n The DeepMind prover agent has found a formal proof of this statement.
"},"MonochromaticQuantumGraph.eqSystem10_no_solution_d3":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem10_no_solution_d3","anchor":"MonochromaticQuantumGraph___eqSystem10_no_solution_d3","docHtml":"\n For $N = 10$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem10_no_solution_d4":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem10_no_solution_d4","anchor":"MonochromaticQuantumGraph___eqSystem10_no_solution_d4","docHtml":"\n For $N = 10$ and $D = 4$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem10_no_solution_d5":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem10_no_solution_d5","anchor":"MonochromaticQuantumGraph___eqSystem10_no_solution_d5","docHtml":"\n For $N = 10$ and $D = 5$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem10_no_solution_d6":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem10_no_solution_d6","anchor":"MonochromaticQuantumGraph___eqSystem10_no_solution_d6","docHtml":"\n For $N = 10$ and $D = 6$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem10_no_solution_d7":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem10_no_solution_d7","anchor":"MonochromaticQuantumGraph___eqSystem10_no_solution_d7","docHtml":"\n For $N = 10$ and $D = 7$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem10_no_solution_d8":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem10_no_solution_d8","anchor":"MonochromaticQuantumGraph___eqSystem10_no_solution_d8","docHtml":"\n For $N = 10$ and $D = 8$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem10_no_solution_d9":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem10_no_solution_d9","anchor":"MonochromaticQuantumGraph___eqSystem10_no_solution_d9","docHtml":"\n For $N = 10$ and $D = 9$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem10_no_solution_d10":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem10_no_solution_d10","anchor":"MonochromaticQuantumGraph___eqSystem10_no_solution_d10","docHtml":"\n For $N = 10$ and $D = 10$, does there exist no solution to the monochromatic quantum\ngraph equation system over $\\mathbb{C}$?
\n\n This follows from the $D = N$ case, see eqSystem_no_solution_even_ge4_d_eq_n_explicit.
\n For $N = 12$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem14_no_solution_d3":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem14_no_solution_d3","anchor":"MonochromaticQuantumGraph___eqSystem14_no_solution_d3","docHtml":"\n For $N = 14$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem16_no_solution_d3":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem16_no_solution_d3","anchor":"MonochromaticQuantumGraph___eqSystem16_no_solution_d3","docHtml":"\n For $N = 16$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem_no_solution_ge6_ge3":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem_no_solution_ge6_ge3","anchor":"MonochromaticQuantumGraph___eqSystem_no_solution_ge6_ge3","docHtml":"\n For all even $N \\geq 6$ and $D \\geq 3$, does there exist no solution to the monochromatic\nquantum graph equation system over $\\mathbb{C}$?
"},"MonochromaticQuantumGraph.eqSystem4_no_solution_ge4_real":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem4_no_solution_ge4_real","anchor":"MonochromaticQuantumGraph___eqSystem4_no_solution_ge4_real","docHtml":"\n For $N = 4$ and all $D \\geq 4$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{R}$?
\n\n This follows from the solution of the complex version of the problem\n(see eqSystem4_no_solution_ge4)
\n For $N = 6$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{R}$?
"},"MonochromaticQuantumGraph.eqSystem6_no_solution_d5_real":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem6_no_solution_d5_real","anchor":"MonochromaticQuantumGraph___eqSystem6_no_solution_d5_real","docHtml":"\n For $N = 6$ and $D = 5$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{R}$?
"},"MonochromaticQuantumGraph.eqSystem6_no_solution_ge3_real":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem6_no_solution_ge3_real","anchor":"MonochromaticQuantumGraph___eqSystem6_no_solution_ge3_real","docHtml":"\n For $N = 6$ and all $D \\geq 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{R}$?
"},"MonochromaticQuantumGraph.eqSystem8_no_solution_d3_real":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem8_no_solution_d3_real","anchor":"MonochromaticQuantumGraph___eqSystem8_no_solution_d3_real","docHtml":"\n For $N = 8$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{R}$?
"},"MonochromaticQuantumGraph.eqSystem10_no_solution_d3_real":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem10_no_solution_d3_real","anchor":"MonochromaticQuantumGraph___eqSystem10_no_solution_d3_real","docHtml":"\n For $N = 10$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{R}$?
"},"MonochromaticQuantumGraph.eqSystem_no_solution_ge6_ge3_real":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem_no_solution_ge6_ge3_real","anchor":"MonochromaticQuantumGraph___eqSystem_no_solution_ge6_ge3_real","docHtml":"\n For all even $N \\geq 6$ and $D \\geq 3$, does there exist no solution to the monochromatic\nquantum graph equation system over $\\mathbb{R}$?
"},"MonochromaticQuantumGraph.eqSystem4_no_solution_ge4_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem4_no_solution_ge4_int","anchor":"MonochromaticQuantumGraph___eqSystem4_no_solution_ge4_int","docHtml":"\n For $N = 4$ and all $D \\geq 4$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$?
\n\n This follows from the solution of the complex version of the problem\n(see eqSystem4_no_solution_ge4).
\n For $N = 6$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$?
"},"MonochromaticQuantumGraph.eqSystem6_no_solution_d5_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem6_no_solution_d5_int","anchor":"MonochromaticQuantumGraph___eqSystem6_no_solution_d5_int","docHtml":"\n For $N = 6$ and $D = 5$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$?
"},"MonochromaticQuantumGraph.eqSystem6_no_solution_ge3_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem6_no_solution_ge3_int","anchor":"MonochromaticQuantumGraph___eqSystem6_no_solution_ge3_int","docHtml":"\n For $N = 6$ and all $D \\geq 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$?
"},"MonochromaticQuantumGraph.eqSystem8_no_solution_d3_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem8_no_solution_d3_int","anchor":"MonochromaticQuantumGraph___eqSystem8_no_solution_d3_int","docHtml":"\n For $N = 8$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$?
"},"MonochromaticQuantumGraph.eqSystem10_no_solution_d3_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem10_no_solution_d3_int","anchor":"MonochromaticQuantumGraph___eqSystem10_no_solution_d3_int","docHtml":"\n For $N = 10$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$?
"},"MonochromaticQuantumGraph.eqSystem_no_solution_ge6_ge3_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem_no_solution_ge6_ge3_int","anchor":"MonochromaticQuantumGraph___eqSystem_no_solution_ge6_ge3_int","docHtml":"\n For all even $N \\geq 6$ and $D \\geq 3$, does there exist no solution to the monochromatic\nquantum graph equation system over $\\mathbb{Z}$?
"},"MonochromaticQuantumGraph.eqSystem4_no_solution_ge4_trinary_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem4_no_solution_ge4_trinary_int","anchor":"MonochromaticQuantumGraph___eqSystem4_no_solution_ge4_trinary_int","docHtml":"\n For $N = 4$ and all $D \\geq 4$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$ with weights in ${-1, 0, 1}$?
\n\n This follows from the complex version, see eqSystem4_no_solution_ge4.
\n For $N = 6$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$ with weights in ${-1, 0, 1}$?
"},"MonochromaticQuantumGraph.eqSystem6_no_solution_d5_trinary_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem6_no_solution_d5_trinary_int","anchor":"MonochromaticQuantumGraph___eqSystem6_no_solution_d5_trinary_int","docHtml":"\n For $N = 6$ and $D = 5$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$ with weights in ${-1, 0, 1}$?
"},"MonochromaticQuantumGraph.eqSystem6_no_solution_ge3_trinary_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem6_no_solution_ge3_trinary_int","anchor":"MonochromaticQuantumGraph___eqSystem6_no_solution_ge3_trinary_int","docHtml":"\n For $N = 6$ and all $D \\geq 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$ with weights in ${-1, 0, 1}$?
"},"MonochromaticQuantumGraph.eqSystem8_no_solution_d3_trinary_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem8_no_solution_d3_trinary_int","anchor":"MonochromaticQuantumGraph___eqSystem8_no_solution_d3_trinary_int","docHtml":"\n For $N = 8$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$ with weights in ${-1, 0, 1}$?
"},"MonochromaticQuantumGraph.eqSystem10_no_solution_d3_trinary_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem10_no_solution_d3_trinary_int","anchor":"MonochromaticQuantumGraph___eqSystem10_no_solution_d3_trinary_int","docHtml":"\n For $N = 10$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$ with weights in ${-1, 0, 1}$?
"},"MonochromaticQuantumGraph.eqSystem_no_solution_ge6_ge3_trinary_int":{"url":"/FormalConjectures/Paper/MonochromaticQuantumGraph/#MonochromaticQuantumGraph___eqSystem_no_solution_ge6_ge3_trinary_int","anchor":"MonochromaticQuantumGraph___eqSystem_no_solution_ge6_ge3_trinary_int","docHtml":"\n For all even $N \\geq 6$ and $D \\geq 3$, does there exist no solution to the monochromatic\nquantum graph equation system over $\\mathbb{Z}$ with weights in ${-1, 0, 1}$?
"},"ClaudesCycles.Vertex":{"url":"/FormalConjectures/Paper/ClaudesCycles/#ClaudesCycles___Vertex","anchor":"ClaudesCycles___Vertex","docHtml":"\n The vertex type: vectors in (ZMod m)³.
\n Bump coordinate b of vertex v: add 1 to the b-th component.
\n Adjacency in the cube digraph: u is adjacent to v if v is obtained from u by\nbumping one coordinate.
\n A permutation σ on vertices is a directed Hamiltonian cycle of a digraph with adjacency\nadj if every arc (v, σ v) is an edge, σ is a single cycle, and σ moves every vertex.
\n The arcs of the cube digraph on (ZMod m)³ can be decomposed into three directed\nHamiltonian cycles: there exist three permutations, each forming a directed Hamiltonian\ncycle, such that every arc (v, bumpAt b v) belongs to exactly one cycle.
\n For odd m > 1, the cube digraph on (ZMod m)³ has a Hamiltonian arc decomposition\ninto three directed cycles [Knu26].
\n The case m = 2 is impossible: the cube digraph on (ZMod 2)³ does not have a\nHamiltonian arc decomposition [Aub82].
\n For even m > 2, it is open whether the cube digraph on (ZMod m)³ has a Hamiltonian\narc decomposition.
\n A polyhedron is Rupert if one can cut a hole in it and pass another\ncopy of the same polyhedron through that hole.
\n\n More formally: a convex body in ℝ³ is a compact, convex set with\nnonempty interior. A convex body X is said to be Rupert if there are\ntwo affine transforms T₁, T₂ ∈ SE(3) such that π(T₁(X)) ⊆\nint(π(T₂(X))), where π : ℝ³ → ℝ² is the evident projection, and int\ndenotes topological interior.
\n\n Not all convex bodies are Rupert. For example,
\n\n the unit ball is not Rupert
\n\n the circular cylinder of unit diameter and height\nclosed on each end by disks is not Rupert
\n\n However, many convex polyhedra are Rupert. All Platonic solids, and\nmost Archimedean and Catalan solids are known to be Rupert.
\n\n Question: are all convex polyhedra with nonempty interior Rupert?
\n\n
\nPlatonic Passages,\nR. P. Jerrard, J. E. Wetzel, and L. Yuan., Math. Mag., 90(2):87–98,\n2017. conjectures (\"with a certain hesitancy\") that perhaps all\nconvex polyhedra are Rupert.
\n\n However, An Algorithmic Approach to Rupert's Problem\ndescribes experimental evidence to suggest that three Archimedean\nsolids may not be Rupert.
\n\nOptimizing for the Rupert property\nis the source of some of the Catalan solid results, and has more\nresults for Johnson polyhedra as well.
\n\nThis video by David Renshaw visualizes\nknown results for Platonic, Archimedean, and Catalan solids.
\n\n This problem's name comes from the fact that it is a generalization\nof Prince Rupert's Cube.
\n\nA convex polyhedron without Rupert's property,\nJakob Steininger and Sergey Yurkevich, 2025. Constructs a convex polyhedron and\na proof that it is not Rupert, resolving the open question.
\n\n The result of transforming a subset of ℝ³ by a chosen rotation and offset,\nand then projected to ℝ².
"},"Rupert.IsRupert":{"url":"/FormalConjectures/Paper/Rupert/#Rupert___IsRupert","anchor":"Rupert___IsRupert","docHtml":"\n A convex polyhedron (given as a finite collection of vertices) is Rupert if\nthere are two rotations in ℝ³ (called \"inner\" and \"outer\") and a translation in ℝ²\nsuch that the \"inner shadow\" (the projection to ℝ² of the inner rotation applied\nto the polyhedron, then translated) fits in the interior of the \"outer shadow\"\n(the projection to ℝ² of the outer rotation applied to the polyhedron)
\n\n [Note: The restriction to (polyhedra determined by the convex hulls of)\n
\n There exists a convex polyhedron with nonempty interior for which the Rupert property does\nnot hold.
"},"VoronovskajaTypeFormula.bernsteinTail":{"url":"/FormalConjectures/Paper/VoronovskajaTypeFormula/#VoronovskajaTypeFormula___bernsteinTail","anchor":"VoronovskajaTypeFormula___bernsteinTail","docHtml":"\n Cumulative sum $J_{n,k}(x) = \\sum_{j=k}^n p_{n,j}(x)$.
"},"VoronovskajaTypeFormula.bezierBernstein":{"url":"/FormalConjectures/Paper/VoronovskajaTypeFormula/#VoronovskajaTypeFormula___bezierBernstein","anchor":"VoronovskajaTypeFormula___bezierBernstein","docHtml":"\n Bézier–type Bernstein operator:\n[\n(B_{n,\\alpha} f)(x)\n= \\sum_{k=0}^{n}\nf!\\left(\\frac{k}{n}\\right)\n\\left(\nJ_{n,k}(x)^{\\alpha}
\n\n J_{n,k+1}(x)^{\\alpha}\n\\right)\n]
\n\n Classical Voronovskaja theorem (α = 1).
\n\n For functions $f$ that are $C^2$ on $[0,1]$, the limit:\n[\nn\\bigl( B_n f(x) - f(x) \\bigr)\n;\\longrightarrow;\n\\frac{1}{2}, x(1 - x), f''(x)\n]
"},"VoronovskajaTypeFormula.voronovskaja_theorem.bezier_bernstein_operators":{"url":"/FormalConjectures/Paper/VoronovskajaTypeFormula/#VoronovskajaTypeFormula___voronovskaja_theorem___bezier_bernstein_operators","anchor":"VoronovskajaTypeFormula___voronovskaja_theorem___bezier_bernstein_operators","docHtml":"\n Conjecture: Voronovskaja-type formula for Bézier-Bernstein operators\nwith shape parameter $\\alpha > 0$, $\\alpha \\neq 1$.
\n\n The source asks for sufficiently smooth functions. This concrete version uses\nContDiffOn ℝ 2 f I as a readable baseline regularity assumption; since the\ndomain is the compact interval $[0,1]$, this also explains why no separate\nboundedness assumption is included here. The variants below record the unknown\nsmoothness threshold more explicitly.
\n Variant of the Bézier-Bernstein Voronovskaja problem which treats \"sufficiently smooth\" as an\neventual condition in the smoothness order $m$: for all sufficiently large finite $m$, every\n$C^m$ function on $[0,1]$ should have the asserted asymptotic formula.
"},"VoronovskajaTypeFormula.voronovskaja_theorem.bezier_bernstein_operators.variants.eventually_smooth.limit_exists":{"url":"/FormalConjectures/Paper/VoronovskajaTypeFormula/#VoronovskajaTypeFormula___voronovskaja_theorem___bezier_bernstein_operators___variants___eventually_smooth___limit_exists","anchor":"VoronovskajaTypeFormula___voronovskaja_theorem___bezier_bernstein_operators___variants___eventually_smooth___limit_exists","docHtml":"\n Existence-only version of the eventual-smoothness variant. This separates the first part of the\nsource problem, proving that the scaled sequence has some limit, from the stronger task of finding\nan explicit expression for that limit.
"},"VoronovskajaTypeFormula.voronovskaja_theorem.bezier_bernstein_operators.variants.answer_smoothness":{"url":"/FormalConjectures/Paper/VoronovskajaTypeFormula/#VoronovskajaTypeFormula___voronovskaja_theorem___bezier_bernstein_operators___variants___answer_smoothness","anchor":"VoronovskajaTypeFormula___voronovskaja_theorem___bezier_bernstein_operators___variants___answer_smoothness","docHtml":"\n Variant of the Bézier-Bernstein Voronovskaja problem with the required smoothness order itself\nleft as an answer. Replacing (answer(sorry) : ℕ × ((ℝ → ℝ) → ℝ → ℝ)) by a concrete value lets one\nstate the conjecture for a chosen regularity threshold.
\n A set $A \\subseteq \\mathbb{R}$ is a union of $n$ intervals if it can be written as\n$A = \\bigcup_{i=1}^{n} (a_i, b_i)$ with $a_1 < b_1 < a_2 < b_2 < \\dots < a_n < b_n$,\ni.e., as a disjoint union of $n$ nondegenerate open intervals in strict order with strict\nseparation. This matches the convention of the paper\n(see e.g. Theorem 3.4 there).
"},"WeakTiling.IsFiniteUnionOfIntervals":{"url":"/FormalConjectures/Paper/WeakTiling/#WeakTiling___IsFiniteUnionOfIntervals","anchor":"WeakTiling___IsFiniteUnionOfIntervals","docHtml":"\n A set $A \\subseteq \\mathbb{R}$ is a finite union of intervals if it is a union of $n$\nintervals for some $n$.
"},"WeakTiling.IsWeakTilingMeasure":{"url":"/FormalConjectures/Paper/WeakTiling/#WeakTiling___IsWeakTilingMeasure","anchor":"WeakTiling___IsWeakTilingMeasure","docHtml":"\n A positive, locally finite Borel measure $\\nu$ on $\\mathbb{R}$ is a weak tiling measure\nfor a bounded measurable set $\\Omega \\subset \\mathbb{R}$ if the convolution\n$1_\\Omega \\ast \\nu = 1_{\\Omega^c}$ holds almost everywhere, i.e.,\n$\\int 1_\\Omega(x - t) , d\\nu(t) = 1_{\\Omega^c}(x)$ for a.e. $x \\in \\mathbb{R}$.
\n\n This is Definition 1.1 from the paper.
"},"WeakTiling.IsProperTiling":{"url":"/FormalConjectures/Paper/WeakTiling/#WeakTiling___IsProperTiling","anchor":"WeakTiling___IsProperTiling","docHtml":"\n A proper tiling of $\\Omega^c$ by translates of $\\Omega$ is specified by a set of\ntranslation parameters $T \\subseteq \\mathbb{R}$ such that the sum of Dirac masses on $T$\nis a weak tiling measure for $\\Omega$.
"},"WeakTiling.HasBoundedDensity":{"url":"/FormalConjectures/Paper/WeakTiling/#WeakTiling___HasBoundedDensity","anchor":"WeakTiling___HasBoundedDensity","docHtml":"\n A set $\\Lambda \\subseteq \\mathbb{R}$ has bounded density if the number of points of\n$\\Lambda$ in any unit open interval is uniformly bounded:\n$\\sup_{x \\in \\mathbb{R}} #(\\Lambda \\cap (x, x + 1)) < \\infty$.
"},"WeakTiling.problem_4_1":{"url":"/FormalConjectures/Paper/WeakTiling/#WeakTiling___problem_4_1","anchor":"WeakTiling___problem_4_1","docHtml":"\nProblem 4.1. Let $\\Omega \\subset \\mathbb{R}$ be a finite union of intervals and $\\nu$\na weak tiling measure for $\\Omega$. Must $\\mathrm{supp}(\\nu)$ have bounded density?
"},"WeakTiling.problem_4_2":{"url":"/FormalConjectures/Paper/WeakTiling/#WeakTiling___problem_4_2","anchor":"WeakTiling___problem_4_2","docHtml":"\nProblem 4.2. Let $\\Omega \\subset \\mathbb{R}$ be a finite union of three or more\nintervals. If $\\Omega$ weakly tiles its complement, must it also tile its complement\nproperly?
"},"WeakTiling.problem_4_3":{"url":"/FormalConjectures/Paper/WeakTiling/#WeakTiling___problem_4_3","anchor":"WeakTiling___problem_4_3","docHtml":"\nProblem 4.3. Let $\\Omega \\subset \\mathbb{R}$ be a finite union of intervals and $\\nu$\na weak tiling measure for $\\Omega$. Must $\\nu$ be expressible as a convex combination of\nproper tiling measures?
"},"Gourevitch.gourevitch_series_identity":{"url":"/FormalConjectures/Paper/Gourevitch/#Gourevitch___gourevitch_series_identity","anchor":"Gourevitch___gourevitch_series_identity","docHtml":"\n The Gourevitch series identity:\nThe following idenitity holds:\n$\\sum_{n=0}^{\\infty} \\frac{1 + 14 n + 76 n^2 + 168 n^3}{2^{20 n}} \\binom{2n}{n}^7 = \\frac{32}{\\pi^3}.$\nThis was originally conjectured in [G2003] by Guillera and proven in [A2025] by Au.
"},"DubnerConjecture.IsTwinPrime":{"url":"/FormalConjectures/Paper/Dubner/#DubnerConjecture___IsTwinPrime","anchor":"DubnerConjecture___IsTwinPrime","docHtml":"\n A twin prime is a prime number that has a prime gap of 2, meaning either p - 2 or p + 2\nis also prime.
"},"DubnerConjecture.t1":{"url":"/FormalConjectures/Paper/Dubner/#DubnerConjecture___t1","anchor":"DubnerConjecture___t1"},"DubnerConjecture.t2":{"url":"/FormalConjectures/Paper/Dubner/#DubnerConjecture___t2","anchor":"DubnerConjecture___t2"},"DubnerConjecture.t3":{"url":"/FormalConjectures/Paper/Dubner/#DubnerConjecture___t3","anchor":"DubnerConjecture___t3"},"DubnerConjecture.t4":{"url":"/FormalConjectures/Paper/Dubner/#DubnerConjecture___t4","anchor":"DubnerConjecture___t4"},"DubnerConjecture.t5":{"url":"/FormalConjectures/Paper/Dubner/#DubnerConjecture___t5","anchor":"DubnerConjecture___t5"},"DubnerConjecture.dubner_conjecture":{"url":"/FormalConjectures/Paper/Dubner/#DubnerConjecture___dubner_conjecture","anchor":"DubnerConjecture___dubner_conjecture","docHtml":"\n Every even number greater than 4208 is the sum of two twin primes.
"},"LatinTableau.SimpleGraph.LatinTableauConjecture":{"url":"/FormalConjectures/Paper/LatinTableau/#LatinTableau___SimpleGraph___LatinTableauConjecture","anchor":"LatinTableau___SimpleGraph___LatinTableauConjecture","docHtml":"\n The Latin Tableau Conjecture: If G is the simple graph\nof a Young diagram, then G is CDS-colorable.
"},"ReedOmegaDeltaChi.reed_omega_delta_chi_conjecture":{"url":"/FormalConjectures/Paper/ReedOmegaDeltaChi/#ReedOmegaDeltaChi___reed_omega_delta_chi_conjecture","anchor":"ReedOmegaDeltaChi___reed_omega_delta_chi_conjecture","docHtml":"\n For a graph $G$, we define $\\Delta(G)$ to be the maximum degree, $\\omega(G)$ to be the size of the\nlargest clique subgraph, and $\\chi(G)$ to be the chromatic number. Reed's omega, delta, and chi\nconjecture states that $$\\chi(G) \\leq \\lceil \\frac{1}{2}(\\omega(G) + \\Delta(G) + 1) \\rceil.$$
"},"ReedOmegaDeltaChi.reed_omega_delta_chi_conjecture_for_finite_graphs":{"url":"/FormalConjectures/Paper/ReedOmegaDeltaChi/#ReedOmegaDeltaChi___reed_omega_delta_chi_conjecture_for_finite_graphs","anchor":"ReedOmegaDeltaChi___reed_omega_delta_chi_conjecture_for_finite_graphs","docHtml":"\n For a finite graph $G$, we define $\\Delta(G)$ to be the maximum degree, $\\omega(G)$ to be the\nsize of the largest clique subgraph, and $\\chi(G)$ to be the chromatic number. Reed's omega,\ndelta, and chi conjecture states that $$\\chi(G) \\leq \\lceil \\frac{1}{2}(\\omega(G) + \\Delta(G) + 1) \\rceil.$$
"},"ReedOmegaDeltaChi.reed_conjecture_Δ_6_ω_2":{"url":"/FormalConjectures/Paper/ReedOmegaDeltaChi/#ReedOmegaDeltaChi___reed_conjecture_____6_____2","anchor":"ReedOmegaDeltaChi___reed_conjecture_____6_____2","docHtml":"\n The simplest open case is when $\\Delta(G) = 6$ and $\\omega(G) = 2$.
"},"StrongSensitivityConjecture.flip":{"url":"/FormalConjectures/Paper/StrongSensitivityConjecture/#StrongSensitivityConjecture___flip","anchor":"StrongSensitivityConjecture___flip","docHtml":"\n Flip operator,\nflip x B returns input x with bits in block B inverted.
\n Local sensitivity s(f,x),\nnumber of indices where flipping one bit changes the value of f.
\n Global sensitivity s(f),\nmaximum sensitivity of f over all inputs.
\n Check validity of block collection (disjoint and sensitive),\nA collection of blocks cB is valid for f at x if the blocks are\ndisjoint and flipping any block changes f(x).
\n Local block sensitivity bs(f,x),\nmaximum size of a collection of sensitive, disjoint blocks for f at x.
\n Global block sensitivity of f,\nmaximum block sensitivity of f over all inputs.
\n Strong Sensitivity Conjecture,\nfor every Boolean function f : {0,1}^n → {0,1},\nbs(f) ≤ s(f)^2.
\n We call this the s(f). Huang's\ncelebrated result (often called the sensitivity theorem) gives a quartic bound,\nbs(f) ≤ s(f)^4, thereby settling the original conjecture.
\n Simple test example,\nA Boolean function whose block sensitivity is strictly greater than\nits sensitivity. Source: Nisan1989.
\n\nnisanExample(x) = 1 iff the Hamming weight of x is either\nn/2 or n/2 + 1. We assume n is a multiple of 4.\nThe function is symmetric, so its value only depends on the Hamming weight\nof the input.
\n Assuming n is a multiple of 4, the sensitivity of nisanExample\nis n/2, achieved by any x with Hamming weight n/2.
\n Assuming n is a multiple of 4, the block sensitivity of nisanExample\nis 3n/4, achieved by any x with Hamming weight n/2.\nAn optimal block configuration uses all n/2 1-bits as singleton blocks\nand forms n/4 disjoint size-2 blocks from the 0-bits.
\n A rational number is fusible if it belongs to the smallest set containing $0$ and closed under\nthe operation\n$$\na \\sim b = \\frac{a + b + 1}{2}\n$$\nwhenever $|a-b| < 1$.
"},"FusibleNumber.IsFusible.zero":{"url":"/FormalConjectures/Paper/FusibleNumber/#FusibleNumber___IsFusible___zero","anchor":"FusibleNumber___IsFusible___zero","docHtml":"\n A rational number is fusible if it belongs to the smallest set containing $0$ and closed under\nthe operation\n$$\na \\sim b = \\frac{a + b + 1}{2}\n$$\nwhenever $|a-b| < 1$.
"},"FusibleNumber.IsFusible.fuse":{"url":"/FormalConjectures/Paper/FusibleNumber/#FusibleNumber___IsFusible___fuse","anchor":"FusibleNumber___IsFusible___fuse","docHtml":"\n A rational number is fusible if it belongs to the smallest set containing $0$ and closed under\nthe operation\n$$\na \\sim b = \\frac{a + b + 1}{2}\n$$\nwhenever $|a-b| < 1$.
"},"FusibleNumber.isFusible_one_half":{"url":"/FormalConjectures/Paper/FusibleNumber/#FusibleNumber___isFusible_one_half","anchor":"FusibleNumber___isFusible_one_half","docHtml":"\n The rational number $1/2$ is fusible.
"},"FusibleNumber.isFusible_one":{"url":"/FormalConjectures/Paper/FusibleNumber/#FusibleNumber___isFusible_one","anchor":"FusibleNumber___isFusible_one","docHtml":"\n The rational number $1$ is fusible.
"},"FusibleNumber.conj_7_1":{"url":"/FormalConjectures/Paper/FusibleNumber/#FusibleNumber___conj_7_1","anchor":"FusibleNumber___conj_7_1","docHtml":"\n If x is a fusible number and y is its successor, then the interval [x + 1, y + 1) can be\ndivided into intervals [ℓₙ, ℓₙ₊₁), such that the fusible numbers in [ℓₙ, ℓₙ₊₁) are obtained by\nfusing the n + 1st successor of x with a fusible number.\nThis formalization differs from Conjecture 7.1 in the paper in four ways:\n(1) it is obtained from Conjecture 7.1 by plugging in n + 1 into n, which simplifies the expressions\nand removes the need to assume n ≥ 1;\n(2) the n + 1st successor s^(n+1)(x) is replaced by the explicit value x + (2 - 1 / 2 ^ n) * m;\n(3) instead of defining y to be the successor of x, we assert that there is no fusible number\nstrictly between x and y;\n(4) instead of using ∃ z, IsFusible z ∧ q = s^(n+1)(x) ~ z we use the value of z determined by the equality,\nnamely z = 2 * q - 1 - s^(n+1)(x), and it is easy to see z ∈ [x + 1 - m / 2 ^ n, x + 1) as required.
\n A topological space $X$ is called
\n For all $x : X, n : ℕ$ we have $x ∈ V x n$
\n\n For all $x : X, n : ℕ$: $V x (n + 1) ⊆ V x n$
\n\n $O ⊆ X$ is open iff $∀ x ∈ O, ∃ n : ℕ, V x n ⊆ O$
\n\n A topological space $X$ is called
\n For all $x : X, n : ℕ$ we have $x ∈ V x n$
\n\n For all $x : X, n : ℕ$: $V x (n + 1) ⊆ V x n$
\n\n $O ⊆ X$ is open iff $∀ x ∈ O, ∃ n : ℕ, V x n ⊆ O$
\n\n There are weakly first countable spaces which are not first countable,\nfor example the Arens Space.
"},"WeaklyFirstCountable.FirstCountableTopology.weaklyFirstCountableTopology":{"url":"/FormalConjectures/Paper/WeaklyFirstCountable/#WeaklyFirstCountable___FirstCountableTopology___weaklyFirstCountableTopology","anchor":"WeaklyFirstCountable___FirstCountableTopology___weaklyFirstCountableTopology","docHtml":"\n Every first countable space is weakly first countable,\nsimply take $N x$ as a countable neighborhood basis of $x$.
"},"WeaklyFirstCountable.existsWeaklyFirstCountableCompactBig":{"url":"/FormalConjectures/Paper/WeaklyFirstCountable/#WeaklyFirstCountable___existsWeaklyFirstCountableCompactBig","anchor":"WeaklyFirstCountable___existsWeaklyFirstCountableCompactBig","docHtml":"\n Problem 2 in [Ar2013]: Give an example in ZFC of a weakly first-\ncountable compact space X such that $𝔠 < |X|$.
"},"WeaklyFirstCountable.ExistsWeaklyFirstCountableCompactNotFirstCountable":{"url":"/FormalConjectures/Paper/WeaklyFirstCountable/#WeaklyFirstCountable___ExistsWeaklyFirstCountableCompactNotFirstCountable","anchor":"WeaklyFirstCountable___ExistsWeaklyFirstCountableCompactNotFirstCountable","docHtml":"\n Problem 3 in [Ar2013]: Give an example in ZFC of a weakly first-\ncountable compact space which is not first countable.
"},"WeaklyFirstCountable.existsWeaklyFirstCountableCompactNotFirstCountable":{"url":"/FormalConjectures/Paper/WeaklyFirstCountable/#WeaklyFirstCountable___existsWeaklyFirstCountableCompactNotFirstCountable","anchor":"WeaklyFirstCountable___existsWeaklyFirstCountableCompactNotFirstCountable","docHtml":"\n Problem 3 in [Ar2013]: Give an example in ZFC of a weakly first-\ncountable compact space which is not first countable.
"},"WeaklyFirstCountable.CH.existsWeaklyFirstCountableCompactNotFirstCountable":{"url":"/FormalConjectures/Paper/WeaklyFirstCountable/#WeaklyFirstCountable___CH___existsWeaklyFirstCountableCompactNotFirstCountable","anchor":"WeaklyFirstCountable___CH___existsWeaklyFirstCountableCompactNotFirstCountable","docHtml":"\n Under CH, such a space exists as constructed in [Ya1976] by Yakovlev.
"},"Chvatal.Decreasing":{"url":"/FormalConjectures/Paper/Chvatal/#Chvatal___Decreasing","anchor":"Chvatal___Decreasing","docHtml":"\n A family F of sets is Decreasing if it is closed under taking subsets.
"},"Chvatal.Intersecting":{"url":"/FormalConjectures/Paper/Chvatal/#Chvatal___Intersecting","anchor":"Chvatal___Intersecting","docHtml":"\n A family F of sets is Intersecting if each pair of members has nonempty intersection.
"},"Chvatal.exists_maximal_star":{"url":"/FormalConjectures/Paper/Chvatal/#Chvatal___exists_maximal_star","anchor":"Chvatal___exists_maximal_star","docHtml":"\n If F is a decreasing family of sets of some finite type α, then there is some element\nx of α such that the family consisting of all members of F containing x is an intersecting\nsubfamily of F with maximal cardinality.
"},"CardinalityLindelof.HasGδSingletons.lindelof_card":{"url":"/FormalConjectures/Paper/CardinalityLindelof/#CardinalityLindelof___HasG___Singletons___lindelof_card","anchor":"CardinalityLindelof___HasG___Singletons___lindelof_card","docHtml":"\n Is there a Lindelöf space with singletons as Gδ sets with cardinality greater than the continuum?
"},"PrimeTuplesConjecture.prime_tuples_conjecture":{"url":"/FormalConjectures/Paper/PrimeTuples/#PrimeTuplesConjecture___prime_tuples_conjecture","anchor":"PrimeTuplesConjecture___prime_tuples_conjecture","docHtml":"\n For any k ≥ 2, let a₁,...,aₖ and b₁,...,bₖ be integers with aᵢ > 0. Suppose that for\nevery prime p there exists an integer n such that p ∤ ∏ i, (aᵢ n + bᵢ). Then there exist\ninfinitely many n such that aᵢ n + bᵢ is prime for all i.
\n A topological space $X$ is called
\n A topological space $X$ is called
\n Every discrete space is homogeneous.
"},"Homogeneous.homogeneousSpace_exists_inj_tendsto":{"url":"/FormalConjectures/Paper/Homogenous/#Homogeneous___homogeneousSpace_exists_inj_tendsto","anchor":"Homogeneous___homogeneousSpace_exists_inj_tendsto","docHtml":"\n Problem 13 in [Ar2013]:\nIs it true that every infinite homogeneous compact\nspace contains a non-trivial convergent sequence?
"},"Homogeneous.homogeneousSpace_exists_surjective":{"url":"/FormalConjectures/Paper/Homogenous/#Homogeneous___homogeneousSpace_exists_surjective","anchor":"Homogeneous___homogeneousSpace_exists_surjective","docHtml":"\n Problem 14 in [Ar2013]:\nIs it possible to represent an arbitrary compact space as an image\nof a homogeneous compact space under a continuous mapping?
"},"Homogeneous.CountablyMonolithicSpace":{"url":"/FormalConjectures/Paper/Homogenous/#Homogeneous___CountablyMonolithicSpace","anchor":"Homogeneous___CountablyMonolithicSpace","docHtml":"\n A topological space is called ω-monolithic if\nthe closure of every countable subspace is metrizable.
"},"Homogeneous.CountablyMonolithicSpace.metrizable_of_closure_of_countable":{"url":"/FormalConjectures/Paper/Homogenous/#Homogeneous___CountablyMonolithicSpace___metrizable_of_closure_of_countable","anchor":"Homogeneous___CountablyMonolithicSpace___metrizable_of_closure_of_countable","docHtml":"\n A topological space is called ω-monolithic if\nthe closure of every countable subspace is metrizable.
"},"Homogeneous.MetrizableSpace.countablyMonolithicSpace":{"url":"/FormalConjectures/Paper/Homogenous/#Homogeneous___MetrizableSpace___countablyMonolithicSpace","anchor":"Homogeneous___MetrizableSpace___countablyMonolithicSpace","docHtml":"\n Every Metrizable space is ω-monolithic.
"},"Homogeneous.firstCountableTopology_of_countablyMonolithicSpace":{"url":"/FormalConjectures/Paper/Homogenous/#Homogeneous___firstCountableTopology_of_countablyMonolithicSpace","anchor":"Homogeneous___firstCountableTopology_of_countablyMonolithicSpace","docHtml":"\n Problem 15 in [Ar2013]:\nIs every homogeneous ω-monolithic compact space first countable?
"},"Homogeneous.countablyMonolithicSpace_card_lt":{"url":"/FormalConjectures/Paper/Homogenous/#Homogeneous___countablyMonolithicSpace_card_lt","anchor":"Homogeneous___countablyMonolithicSpace_card_lt","docHtml":"\n Problem 16 in [Ar2013]:\nIs the cardinality of every homogeneous ω-monolithic compact space not greater than 𝔠?
"},"Homogeneous.countablyMonolithicSpace_exists_nhds_generated_countable":{"url":"/FormalConjectures/Paper/Homogenous/#Homogeneous___countablyMonolithicSpace_exists_nhds_generated_countable","anchor":"Homogeneous___countablyMonolithicSpace_exists_nhds_generated_countable","docHtml":"\n Problem 17 in [Ar2013]:\nIs it true that every nonempty ω-monolithic compact space contains a point with a\nfirst countable neighborhood basis?
\n\n Note: Nonempty X is required since the conclusion asserts the existence of a point.
\n Conjecture 3.2 in [Wa2011]:\nEach Latin square of odd order has at least one transversal.
"},"LatinSquare.oddOrderLeq9LatinSquareTransversal":{"url":"/FormalConjectures/Paper/LatinSquare/#LatinSquare___oddOrderLeq9LatinSquareTransversal","anchor":"LatinSquare___oddOrderLeq9LatinSquareTransversal","docHtml":"\n The conjecture is known to be true for $n \\leq 9$.
"},"LatinSquare.latinSquareOrder11Transversal":{"url":"/FormalConjectures/Paper/LatinSquare/#LatinSquare___latinSquareOrder11Transversal","anchor":"LatinSquare___latinSquareOrder11Transversal","docHtml":"\n The smallest odd number for which this conjecture is not known is 11.
"},"LatinSquare.latinSquareNearTransversal":{"url":"/FormalConjectures/Paper/LatinSquare/#LatinSquare___latinSquareNearTransversal","anchor":"LatinSquare___latinSquareNearTransversal","docHtml":"\n Conjecture 5.1 in [Wa2011]:\nEvery latin square has a near-transversal
"},"LatinSquare.z":{"url":"/FormalConjectures/Paper/LatinSquare/#LatinSquare___z","anchor":"LatinSquare___z","docHtml":"\n The number of transversals of the Cayley table of the cyclic group $\\mathbb{Z}_n$
"},"LatinSquare.z_zero":{"url":"/FormalConjectures/Paper/LatinSquare/#LatinSquare___z_zero","anchor":"LatinSquare___z_zero","docHtml":"\n The $0 \\times 0$ Cayley table has exactly $1$ transversal (vacuously).
"},"LatinSquare.z_odd_values":{"url":"/FormalConjectures/Paper/LatinSquare/#LatinSquare___z_odd_values","anchor":"LatinSquare___z_odd_values","docHtml":"\n The number of transversals of the Cayley table of $\\mathbb{Z}_n$ for odd $n$ forms\nOEIS A006717, starting with\n$z(1) = 1, z(3) = 3, z(5) = 15, z(7) = 133$.
"},"LatinSquare.z_even":{"url":"/FormalConjectures/Paper/LatinSquare/#LatinSquare___z_even","anchor":"LatinSquare___z_even","docHtml":"\n The Cayley table of $\\mathbb{Z}_n$ for positive even $n$ has no transversals.
"},"LatinSquare.numTransversalsZn":{"url":"/FormalConjectures/Paper/LatinSquare/#LatinSquare___numTransversalsZn","anchor":"LatinSquare___numTransversalsZn","docHtml":"\n Conjecture 6.7 in [Wa2011]:\nThere exist real constants $0 < c_1 < c_2 < 1$ such that\n$$\nc_1^n n! \\leq z_n \\leq c_2^n n!\n$$\nfor all odd $n \\geq 3$.
"},"LatinSquare.growthRateZn":{"url":"/FormalConjectures/Paper/LatinSquare/#LatinSquare___growthRateZn","anchor":"LatinSquare___growthRateZn","docHtml":"\n Conjecture 6.9 in [Wa2011]:\n$$\n\\lim_{n \\to \\infty} \\frac{1}{n} \\log(z_n / n!) = -1\n$$\nIt is not even known if this limit exists.
"},"LatinSquare.T":{"url":"/FormalConjectures/Paper/LatinSquare/#LatinSquare___T","anchor":"LatinSquare___T","docHtml":"\n The maximum number of transversals over all latin squares of order n.
\n Theorem 7.2 in [Wa2011]:\nFor all $n \\geq 5$,\n$$\n15^{n/5} \\leq T(n) \\leq c^n \\sqrt{n} \\cdot n!\n$$\nwhere $c = \\sqrt{\\frac{3 - \\sqrt{3}}{6}} \\cdot e^{\\sqrt{3}/6}$
"},"ZagierMZV.multiZeta":{"url":"/FormalConjectures/Paper/ZagierMZV/#ZagierMZV___multiZeta","anchor":"ZagierMZV___multiZeta","docHtml":"\n The multiple zeta value $\\zeta(s_1, s_2, \\ldots, s_k)$, defined as\n$$\\zeta(s_1, \\ldots, s_k) = \\sum_{n_1 > n_2 > \\cdots > n_k > 0}\n\\frac{1}{n_1^{s_1} n_2^{s_2} \\cdots n_k^{s_k}}.$$\nThe argument is a list of positive natural numbers. The value is well-defined (i.e. the series\nconverges) when the first entry is at least 2, but we define it for all inputs.\nFor the empty list, multiZeta [] = 1 (the empty product convention).
\n Auxiliary function for multiZeta: computes the inner sum\n$\\sum_{n_2 > \\cdots > n_k > 0, n_2 < \\text{bound}} \\frac{1}{n_2^{s_2} \\cdots n_k^{s_k}}$.
\n The
\n An MZV index is
\n The set of all MZV values of weight $n$.
"},"ZagierMZV.mzvSpanOfWeight":{"url":"/FormalConjectures/Paper/ZagierMZV/#ZagierMZV___mzvSpanOfWeight","anchor":"ZagierMZV___mzvSpanOfWeight","docHtml":"\n The $\\mathbb{Q}$-submodule of $\\mathbb{R}$ spanned by all MZVs of weight $n$.
"},"ZagierMZV.zagierDim":{"url":"/FormalConjectures/Paper/ZagierMZV/#ZagierMZV___zagierDim","anchor":"ZagierMZV___zagierDim","docHtml":"\n The conjectured Zagier dimension sequence $d_n$, defined by $d_0 = 1$, $d_1 = 0$, $d_2 = 1$,\nand $d_n = d_{n-2} + d_{n-3}$ for $n \\geq 3$.
"},"ZagierMZV.zagier_conjecture":{"url":"/FormalConjectures/Paper/ZagierMZV/#ZagierMZV___zagier_conjecture","anchor":"ZagierMZV___zagier_conjecture","docHtml":"\nZagier's conjecture
\n\n The $\\mathbb{Q}$-dimension of the vector space spanned by all multiple zeta values\nof weight $n$ equals $d_n$, where $d_n$ is the Zagier dimension sequence\nsatisfying $d_0 = 1$, $d_1 = 0$, $d_2 = 1$, and $d_n = d_{n-2} + d_{n-3}$ for $n \\geq 3$.
"},"ZagierMZV.zagier_upper_bound":{"url":"/FormalConjectures/Paper/ZagierMZV/#ZagierMZV___zagier_upper_bound","anchor":"ZagierMZV___zagier_upper_bound","docHtml":"\nUpper bound [Te02, DG05]
\n\n The dimension of the $\\mathbb{Q}$-vector space of MZVs of weight $n$ is at most $d_n$.
"},"ZagierMZV.zagierDim_first_values":{"url":"/FormalConjectures/Paper/ZagierMZV/#ZagierMZV___zagierDim_first_values","anchor":"ZagierMZV___zagierDim_first_values","docHtml":"\n The first few values of $d_n$ are $1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, \\ldots$
"},"ZagierMZV.no_admissible_weight_one":{"url":"/FormalConjectures/Paper/ZagierMZV/#ZagierMZV___no_admissible_weight_one","anchor":"ZagierMZV___no_admissible_weight_one","docHtml":"\n There is no admissible index of weight 1 (since $s_1 \\geq 2$ is required).
"},"ZagierMZV.dim_mzv_weight_zero":{"url":"/FormalConjectures/Paper/ZagierMZV/#ZagierMZV___dim_mzv_weight_zero","anchor":"ZagierMZV___dim_mzv_weight_zero","docHtml":"\n $\\mathcal{Z}
\n $\\mathcal{Z}
\n Euler's identity: $\\zeta(2) = \\pi^2/6$.
"},"ZagierMZV.multiZeta_four":{"url":"/FormalConjectures/Paper/ZagierMZV/#ZagierMZV___multiZeta_four","anchor":"ZagierMZV___multiZeta_four","docHtml":"\n Euler's identity for $\\zeta(4) = \\pi^4/90$.
"},"Kurepa.left_factorial":{"url":"/FormalConjectures/Paper/Kurepa/#Kurepa___left_factorial","anchor":"Kurepa___left_factorial","docHtml":"\n Left factorial of n\n$$!n = 0! + 1! + 2! + \\dots + (n-1)!$$
"},"Kurepa.kurepa_conjecture":{"url":"/FormalConjectures/Paper/Kurepa/#Kurepa___kurepa_conjecture","anchor":"Kurepa___kurepa_conjecture","docHtml":"\n For all $n$, $$!n\\not\\equiv 0 \\mod n$$
\n\n This appears as B44 \"Sums of factorials.\"\nin Unsolved Problems in Number Theory\nby
\n This statement can be reduced to the prime case only.
"},"Kurepa.kurepa_conjecture.prime_reduction":{"url":"/FormalConjectures/Paper/Kurepa/#Kurepa___kurepa_conjecture___prime_reduction","anchor":"Kurepa___kurepa_conjecture___prime_reduction","docHtml":"\n Kurepa's conjecture for all integers greater than 2 is equivalent to the conjecture restricted to primes greater than 2.
"},"Kurepa.kurepa_conjecture.variants.gcd":{"url":"/FormalConjectures/Paper/Kurepa/#Kurepa___kurepa_conjecture___variants___gcd","anchor":"Kurepa___kurepa_conjecture___variants___gcd","docHtml":"\n An equivalent formulation in terms of the gcd of $n!$ and $!n$.
"},"Kurepa.kurepa_conjecture.gcd_reduction":{"url":"/FormalConjectures/Paper/Kurepa/#Kurepa___kurepa_conjecture___gcd_reduction","anchor":"Kurepa___kurepa_conjecture___gcd_reduction","docHtml":"\n Kurepa's conjecture for all integers greater than 2 is equivalent to the statement that $\\gcd(n!, !n) = 2$ for all integers greater than 2.
"},"Kurepa.kurepa_conjecture.variants.first_cases":{"url":"/FormalConjectures/Paper/Kurepa/#Kurepa___kurepa_conjecture___variants___first_cases","anchor":"Kurepa___kurepa_conjecture___variants___first_cases","docHtml":"\n Sanity check: for small values we can just compute that the conjecture is true
"},"Kurepa.kurepa_conjecture.variants.gcd.first_cases":{"url":"/FormalConjectures/Paper/Kurepa/#Kurepa___kurepa_conjecture___variants___gcd___first_cases","anchor":"Kurepa___kurepa_conjecture___variants___gcd___first_cases","docHtml":"\n Sanity check: for small values we can just compute that the conjecture is true.
"},"CasasAlvero.HasCasasAlveroProp":{"url":"/FormalConjectures/Paper/CasasAlvero/#CasasAlvero___HasCasasAlveroProp","anchor":"CasasAlvero___HasCasasAlveroProp","docHtml":"\n A polynomial P satisfies the Casas-Alvero property if it shares a factor with each\nof its Hasse derivatives up to order d-1, where d is the degree of P.
\n A stronger version of the Casas-Alvero property, which requires that the polynomial P\nshares a root with each of its Hasse derivatives up to order deg P - 1.\nThe subscript r indicates \"root\" in the definition.
\n Note that whether we use HasCasasAlveroProp or HasCasasAlveroPropᵣ to state the Casas-Alvero conjecture,\nwe obtain the following equivalent statements.
\n The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P\nhas the Casas-Alvero property, then P = (X - α)ᵈ for some α.
\n The Casas-Alvero conjecture holds for polynomials of prime power degree.\nThis was proved by Graf von Bothmer, Labs, Schicho, and van de Woestijne.
\n\n Reference: The Casas-Alvero conjecture for infinitely many degrees
"},"CasasAlvero.casas_alvero.double_prime_power":{"url":"/FormalConjectures/Paper/CasasAlvero/#CasasAlvero___casas_alvero___double_prime_power","anchor":"CasasAlvero___casas_alvero___double_prime_power","docHtml":"\n The Casas-Alvero conjecture holds for polynomials of degree 2p^k where p is prime.\nThis was proved by Graf von Bothmer, Labs, Schicho, and van de Woestijne.
\n Reference: The Casas-Alvero conjecture for infinitely many degrees
"},"CasasAlvero.casas_alvero.positive_char_counterexample":{"url":"/FormalConjectures/Paper/CasasAlvero/#CasasAlvero___casas_alvero___positive_char_counterexample","anchor":"CasasAlvero___casas_alvero___positive_char_counterexample","docHtml":"\n The Casas-Alvero conjecture fails in positive characteristic p for polynomials of degree p + 1.\nThis was shown by Graf von Bothmer, Labs, Schicho, and van de Woestijne.
\n Reference: The Casas-Alvero conjecture for infinitely many degrees
\n\n Formal proof linked here provided by AlphaProof.
"},"HartshorneConjecture.AlgebraicGeometry.Scheme.VectorBundles":{"url":"/FormalConjectures/Paper/HartshorneConjecture/#HartshorneConjecture___AlgebraicGeometry___Scheme___VectorBundles","anchor":"HartshorneConjecture___AlgebraicGeometry___Scheme___VectorBundles","docHtml":"\n A vector bundle over a scheme S is a locally free $\\mathcal{O}_S$-module of finite rank.
\n A vector bundle over a scheme S is a locally free $\\mathcal{O}_S$-module of finite rank.
\n A vector bundle over a scheme S is a locally free $\\mathcal{O}_S$-module of finite rank.
\n A vector bundle over a scheme S is a locally free $\\mathcal{O}_S$-module of finite rank.
\n Vector bundles form a category.
"},"HartshorneConjecture.AlgebraicGeometry.Scheme.hasFiniteCoproductsVectorBundles":{"url":"/FormalConjectures/Paper/HartshorneConjecture/#HartshorneConjecture___AlgebraicGeometry___Scheme___hasFiniteCoproductsVectorBundles","anchor":"HartshorneConjecture___AlgebraicGeometry___Scheme___hasFiniteCoproductsVectorBundles"},"HartshorneConjecture.AlgebraicGeometry.Scheme.VectorBundles.Splitting":{"url":"/FormalConjectures/Paper/HartshorneConjecture/#HartshorneConjecture___AlgebraicGeometry___Scheme___VectorBundles___Splitting","anchor":"HartshorneConjecture___AlgebraicGeometry___Scheme___VectorBundles___Splitting","docHtml":"\n A splitting of a vector bundle 𝓕 is a non-trivial direct sum decomposition of 𝓕
\n A splitting of a vector bundle 𝓕 is a non-trivial direct sum decomposition of 𝓕
\n A splitting of a vector bundle 𝓕 is a non-trivial direct sum decomposition of 𝓕
\n A splitting of a vector bundle 𝓕 is a non-trivial direct sum decomposition of 𝓕
\n There are no indecomposable vector bundles of rank 2 on $\\mathbb{P}^n$ for $n \\ge 7$.\nThis is Conjecture 6.3 in [Har1974].
"},"CatchUp.Player":{"url":"/FormalConjectures/Paper/CatchUpConjecture/#CatchUp___Player","anchor":"CatchUp___Player","docHtml":"\n An arbitrary two elements type indexing the players in the Catch-Up game.
"},"CatchUp.Player.p1":{"url":"/FormalConjectures/Paper/CatchUpConjecture/#CatchUp___Player___p1","anchor":"CatchUp___Player___p1","docHtml":"\n An arbitrary two elements type indexing the players in the Catch-Up game.
"},"CatchUp.Player.p2":{"url":"/FormalConjectures/Paper/CatchUpConjecture/#CatchUp___Player___p2","anchor":"CatchUp___Player___p2","docHtml":"\n An arbitrary two elements type indexing the players in the Catch-Up game.
"},"CatchUp.Player.other":{"url":"/FormalConjectures/Paper/CatchUpConjecture/#CatchUp___Player___other","anchor":"CatchUp___Player___other","docHtml":"\n Returns the other player.
"},"CatchUp.Outcome":{"url":"/FormalConjectures/Paper/CatchUpConjecture/#CatchUp___Outcome","anchor":"CatchUp___Outcome","docHtml":"\n The possible outcomes of a Catch-Up game.
"},"CatchUp.Outcome.win":{"url":"/FormalConjectures/Paper/CatchUpConjecture/#CatchUp___Outcome___win","anchor":"CatchUp___Outcome___win","docHtml":"\n The possible outcomes of a Catch-Up game.
"},"CatchUp.Outcome.loss":{"url":"/FormalConjectures/Paper/CatchUpConjecture/#CatchUp___Outcome___loss","anchor":"CatchUp___Outcome___loss","docHtml":"\n The possible outcomes of a Catch-Up game.
"},"CatchUp.Outcome.draw":{"url":"/FormalConjectures/Paper/CatchUpConjecture/#CatchUp___Outcome___draw","anchor":"CatchUp___Outcome___draw","docHtml":"\n The possible outcomes of a Catch-Up game.
"},"CatchUp.Outcome.neg":{"url":"/FormalConjectures/Paper/CatchUpConjecture/#CatchUp___Outcome___neg","anchor":"CatchUp___Outcome___neg","docHtml":"\n Negates an outcome, swapping win and loss. Used when switching player perspectives.
"},"CatchUp.Outcome.best":{"url":"/FormalConjectures/Paper/CatchUpConjecture/#CatchUp___Outcome___best","anchor":"CatchUp___Outcome___best","docHtml":"\n Computes the best outcome for the current player from a list of possible outcomes.\nWin > Draw > Loss.
"},"CatchUp.valueAux":{"url":"/FormalConjectures/Paper/CatchUpConjecture/#CatchUp___valueAux","anchor":"CatchUp___valueAux","docHtml":"\nvalue remaining s_me s_opp isFirstMove evaluates the position where:
\nremaining is the set S' of numbers not yet taken.
\ns_me is the current score of the player who is about to act (the “current player”).
\ns_opp is the opponent’s current score.
\nisFirstMove indicates whether we are in the special very first move of the whole game,\nwhere the rules force exactly one pick and then the turn passes.
\n The result is the game-theoretic value from the current player’s point of view:\n.win / .loss / .draw, assuming optimal play from both sides.
\n We model a single $x_1,...,x_k$) by recursion:\nthe current player chooses one number x; if they are still strictly behind after taking it, they must\ncontinue the same turn (so the recursive call keeps the same “current player”);\nonce they catch up (score ≥ opponent), the turn ends and we swap players.
\n Public API: The game-theoretic value of Catch-Up on the set S, assuming optimal play.\nReturns .win if player 1 wins, .loss if player 2 wins, .draw if the game is tied.
\n Let (T_N = \\sum_{k=1}^{N} k = \\frac{N(N+1)}{2}).\nIf (T_N) is even (equivalently (N \\equiv 0 \\pmod 4) or (N \\equiv 3 \\pmod 4)),\nthen under optimal play the game Catch-Up(\\(\\{1, \\ldots, N\\}\\)) ends in a draw.
\n A sequence of natural numbers is compact on a set S if consecutive terms at distance\n2 differ by 1 for all k ∈ S.
\n The number of vertices of G having degree d.
\nLemma 1 (a)\nIf a sequence d is nondecreasing and no three terms are equal, then terms at distance 2 differ by at least 1.
\nLemma 1 (b)\nIf a sequence d is nondecreasing and no three terms are equal, then terms at distance 2 * r differ by at least r.
\nLemma 2 (a)\nInequality involving sums of terms of a nondecreasing sequence with no three terms equal.
"},"DegreeSequencesTriangleFree.lemma2_b":{"url":"/FormalConjectures/Paper/DegreeSequencesTriangleFree/#DegreeSequencesTriangleFree___lemma2_b","anchor":"DegreeSequencesTriangleFree___lemma2_b","docHtml":"\nLemma 2 (b)\nInequality involving sums of terms of a nondecreasing sequence with no three terms equal.
"},"DegreeSequencesTriangleFree.lemma2_c":{"url":"/FormalConjectures/Paper/DegreeSequencesTriangleFree/#DegreeSequencesTriangleFree___lemma2_c","anchor":"DegreeSequencesTriangleFree___lemma2_c","docHtml":"\nLemma 2 (c)\nInequality involving sums of terms of a nondecreasing sequence with no three terms equal.
"},"DegreeSequencesTriangleFree.lemma2_d":{"url":"/FormalConjectures/Paper/DegreeSequencesTriangleFree/#DegreeSequencesTriangleFree___lemma2_d","anchor":"DegreeSequencesTriangleFree___lemma2_d","docHtml":"\nLemma 2 (d)\nInequality involving sums of terms of a nondecreasing sequence with no three terms equal.
"},"SimpleGraph.HasCompactdegreeSequence":{"url":"/FormalConjectures/Paper/DegreeSequencesTriangleFree/#SimpleGraph___HasCompactdegreeSequence","anchor":"SimpleGraph___HasCompactdegreeSequence","docHtml":"\n The degree sequence of G is compact if it satisfies\nIsCompactSequenceOn for all valid indices k such that k + 2 < Fintype.card α.
\nTheorem 1. If a triangle-free graph has f = 2,\nthen it is bipartite, has minimum degree 1, and\nits degree sequence is compact.
\nLemma 3. For every n there exists a bipartite graph with\n8 n vertices, minimum degree n + 1, and f = 3.
\nLemma 4. Let G be a triangle-free graph with n vertices and let v be a vertex of G.\nThere exists a triangle-free graph H containing G as an induced subgraph such that:\n(i) the degree of v in H is one more than its degree in G;\n(ii) for every vertex w of G other than v the degree of w in H is the same as its degree in G;\n(iii) if J is the subgraph of H induced by the vertices not in G, then f(J)=3 and δ(J) ≥ 2n.
\nTheorem 2. Every triangle-free graph is an induced subgraph of one\nwith f = 3.
\nF n is the smallest number of vertices of a triangle-free graph\nwith chromatic number n and f = 3.
\n The smallest number of vertices of a triangle-free graph with chromatic number 3 and f=3 is 7.
"},"SimpleGraph.F_four_le":{"url":"/FormalConjectures/Paper/DegreeSequencesTriangleFree/#SimpleGraph___F_four_le","anchor":"SimpleGraph___F_four_le","docHtml":"\n The smallest number of vertices of a triangle-free graph with chromatic number 4 and f=3 is at most 19.
"},"OeisA287616.IsSumOfTriangularAndGeneralizedPolygonal":{"url":"/FormalConjectures/OEIS/«287616»/#OeisA287616___IsSumOfTriangularAndGeneralizedPolygonal","anchor":"OeisA287616___IsSumOfTriangularAndGeneralizedPolygonal","docHtml":"\n The predicate that n can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ for\nnonnegative integers $x, y, z$.
\nZhi-Wei Sun's Conjecture (A287616): Any nonnegative integer can be written as the sum of\na triangular number $x(x+1)/2$, a generalized pentagonal number $y(3y+1)/2$, and a generalized\nheptagonal number $z(5z+1)/2$, where $x, y, z$ are nonnegative integers.
"},"OeisA80170.GCDCondition":{"url":"/FormalConjectures/OEIS/«80170»/#OeisA80170___GCDCondition","anchor":"OeisA80170___GCDCondition","docHtml":"\n The gcd of the binomial coefficients\n$\\binom{2k}{k}, \\binom{3k}{k}, \\dots, \\binom{(k+1)k}{k} = 1$.
"},"OeisA80170.PrimePowerCondition":{"url":"/FormalConjectures/OEIS/«80170»/#OeisA80170___PrimePowerCondition","anchor":"OeisA80170___PrimePowerCondition","docHtml":"\n Let P be the largest prime power dividing k.\nThen $k / P > P$.
\n Conjecture: The gcd condition is equivalent to the prime power condition.
"},"OeisA280831.HasSquareCondition":{"url":"/FormalConjectures/OEIS/«280831»/#OeisA280831___HasSquareCondition","anchor":"OeisA280831___HasSquareCondition","docHtml":"\n The predicate that n can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative\nintegers such that $x^4 + 1680 y^3 z$ is a square.
\nZhi-Wei Sun's 1680-Conjecture (A280831): Any nonnegative integer can be written as\n$x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square.
"},"OeisA6697.morphism":{"url":"/FormalConjectures/OEIS/«6697»/#OeisA6697___morphism","anchor":"OeisA6697___morphism","docHtml":"\n The morphism σ on {a, b} defined by a ↦ aab, b ↦ b, represented on Bool where\nfalse = a and true = b.
\n The n-th iterate of the morphism applied to [a].
"},"OeisA6697.count_false_morphism":{"url":"/FormalConjectures/OEIS/«6697»/#OeisA6697___count_false_morphism","anchor":"OeisA6697___count_false_morphism"},"OeisA6697.count_true_morphism":{"url":"/FormalConjectures/OEIS/«6697»/#OeisA6697___count_true_morphism","anchor":"OeisA6697___count_true_morphism"},"OeisA6697.count_false_finiteWord":{"url":"/FormalConjectures/OEIS/«6697»/#OeisA6697___count_false_finiteWord","anchor":"OeisA6697___count_false_finiteWord"},"OeisA6697.count_true_finiteWord":{"url":"/FormalConjectures/OEIS/«6697»/#OeisA6697___count_true_finiteWord","anchor":"OeisA6697___count_true_finiteWord"},"OeisA6697.length_finiteWord":{"url":"/FormalConjectures/OEIS/«6697»/#OeisA6697___length_finiteWord","anchor":"OeisA6697___length_finiteWord"},"OeisA6697.infiniteWord":{"url":"/FormalConjectures/OEIS/«6697»/#OeisA6697___infiniteWord","anchor":"OeisA6697___infiniteWord","docHtml":"\n The infinite word w is the fixed point of the morphism starting from 'a'.\nWe define it as the limit: w(i) is the i-th symbol, which stabilizes after\nsufficiently many iterations.
"},"OeisA6697.subwordAt":{"url":"/FormalConjectures/OEIS/«6697»/#OeisA6697___subwordAt","anchor":"OeisA6697___subwordAt","docHtml":"\n A subword (factor) of length n starting at position i.
"},"OeisA6697.subwordsOfLength":{"url":"/FormalConjectures/OEIS/«6697»/#OeisA6697___subwordsOfLength","anchor":"OeisA6697___subwordsOfLength","docHtml":"\n The set of all distinct subwords of length n in the infinite word.
"},"OeisA6697.a":{"url":"/FormalConjectures/OEIS/«6697»/#OeisA6697___a","anchor":"OeisA6697___a","docHtml":"\n The subword complexity function: a(n) is the number of distinct subwords of length n.
"},"OeisA6697.conjecture":{"url":"/FormalConjectures/OEIS/«6697»/#OeisA6697___conjecture","anchor":"OeisA6697___conjecture","docHtml":"\nConjecture (A6697): The generating function for the number of subwords of length $n$\nin the infinite word generated by $a \\mapsto aab, b \\mapsto b$ is\n$$\\sum_{n \\geq 0} a_n x^n = 1 + \\frac{1}{1-x} + \\frac{1}{(1-x)^2}\\left(\\frac{1}{1-x} -\n\\sum_{k \\geq 0} x^{2^{k+1} + k}\\right).$$
\n\n Equivalently, a(n) equals the n-th coefficient of this generating function.
\n\n However, the reference quoted in the OEIS sequence
\n\n J.-P. Allouche and J. Shallit, \"On the subword complexity of the fixed point of a → aab, b → b,\nand generalizations,\" arXiv:1605.02361 [math.CO], 2016.
\n
\n provides an explicit formula\n$$ a_n = \\sum_{i=0}^{n} \\min(2^i,n-i+1). $$
\n\n If one takes this as a definition of a(n) instead,\nit becomes straightforward to prove the conjecture.\nSee https://github.com/AxiomMath/gdm-formal-conjectures/blob/main/docs/OeisA6697.md\nfor a formal proof of the generating function using this definition.
\n\n Hence, a formalization of arXiv:1605.02361\nwould complete a formal proof as below.
"},"OeisA306477.IsSumOfBinomials":{"url":"/FormalConjectures/OEIS/«306477»/#OeisA306477___IsSumOfBinomials","anchor":"OeisA306477___IsSumOfBinomials","docHtml":"\n The predicate that n can be written as $\\binom{w+2}{2} + \\binom{x+3}{4} + \\binom{y+5}{6} + \\binom{z+7}{8}$\nfor nonnegative integers $w, x, y, z$.
\nZhi-Wei Sun's 2-4-6-8 Conjecture (A306477): Any integer $n > 0$ can be written as\n$\\binom{w+2}{2} + \\binom{x+3}{4} + \\binom{y+5}{6} + \\binom{z+7}{8}$ for nonnegative integers $w, x, y, z$.
"},"OeisA34693.a":{"url":"/FormalConjectures/OEIS/«34693»/#OeisA34693___a","anchor":"OeisA34693___a","docHtml":"\n Smallest number $k$ such that $kn + 1$ is prime.
"},"OeisA34693.zero":{"url":"/FormalConjectures/OEIS/«34693»/#OeisA34693___zero","anchor":"OeisA34693___zero"},"OeisA34693.one":{"url":"/FormalConjectures/OEIS/«34693»/#OeisA34693___one","anchor":"OeisA34693___one"},"OeisA34693.two":{"url":"/FormalConjectures/OEIS/«34693»/#OeisA34693___two","anchor":"OeisA34693___two"},"OeisA34693.three":{"url":"/FormalConjectures/OEIS/«34693»/#OeisA34693___three","anchor":"OeisA34693___three"},"OeisA34693.seven":{"url":"/FormalConjectures/OEIS/«34693»/#OeisA34693___seven","anchor":"OeisA34693___seven"},"OeisA34693.exists_k":{"url":"/FormalConjectures/OEIS/«34693»/#OeisA34693___exists_k","anchor":"OeisA34693___exists_k","docHtml":"\n Conjecture: for every $n > 1$ there exists a number $k < n$ such that $nk + 1$ is a prime.
"},"OeisA34693.exists_k_stronger":{"url":"/FormalConjectures/OEIS/«34693»/#OeisA34693___exists_k_stronger","anchor":"OeisA34693___exists_k_stronger","docHtml":"\n A stronger conjecture: for every n there exists a number $k < 1 + n^{0.75}$ such that\n$nk + 1$ is a prime.
"},"OeisA34693.exists_k_best_possible":{"url":"/FormalConjectures/OEIS/«34693»/#OeisA34693___exists_k_best_possible","anchor":"OeisA34693___exists_k_best_possible","docHtml":"\n The expression $1 + n^{0.74}$ does not work as an upper bound.
"},"OeisA34693.a_isBigO":{"url":"/FormalConjectures/OEIS/«34693»/#OeisA34693___a_isBigO","anchor":"OeisA34693___a_isBigO","docHtml":"\n Conjecture: $a(n) = O(\\log(n)\\log(\\log(n)))$.
"},"OeisA34693.a_unbounded":{"url":"/FormalConjectures/OEIS/«34693»/#OeisA34693___a_unbounded","anchor":"OeisA34693___a_unbounded","docHtml":"\n Counter-conjecture to a_isBigO: $a(n) / (\\log n \\log \\log n)$ is unbounded.
\nZhi-Wei Sun's Conjecture (A239957): Every prime $p$ has a primitive root $0 < g < p$ of the\nform $k^2 + 1$, where $k$ is an integer.
"},"OeisA228828.a":{"url":"/FormalConjectures/OEIS/«228828»/#OeisA228828___a","anchor":"OeisA228828___a","docHtml":"\n Numbers n such that $n^2 + \\pi(n)$ is prime.
"},"OeisA228828.a_zero":{"url":"/FormalConjectures/OEIS/«228828»/#OeisA228828___a_zero","anchor":"OeisA228828___a_zero"},"OeisA228828.a_one":{"url":"/FormalConjectures/OEIS/«228828»/#OeisA228828___a_one","anchor":"OeisA228828___a_one"},"OeisA228828.a_two":{"url":"/FormalConjectures/OEIS/«228828»/#OeisA228828___a_two","anchor":"OeisA228828___a_two"},"OeisA228828.a.infinite":{"url":"/FormalConjectures/OEIS/«228828»/#OeisA228828___a___infinite","anchor":"OeisA228828___a___infinite","docHtml":"\n Conjecture: the sequence A228828 is infinite.
"},"OeisA63880.unitaryDivisors":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___unitaryDivisors","anchor":"OeisA63880___unitaryDivisors","docHtml":"\n The set of unitary divisors of $n$: divisors $d$ such that $\\gcd(d, n/d) = 1$.
"},"OeisA63880.usigma":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___usigma","anchor":"OeisA63880___usigma","docHtml":"\n The sum of unitary divisors of $n$, denoted $\\text{usigma}(n)$.
"},"OeisA63880.a":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___a","anchor":"OeisA63880___a","docHtml":"\n A number $n$ is in the sequence A063880 if $\\sigma(n) = 2 \\cdot \\text{usigma}(n)$.
"},"OeisA63880.A":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___A","anchor":"OeisA63880___A","docHtml":"\n The set of numbers in the sequence A063880.
"},"OeisA63880.isPrimitiveTerm":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___isPrimitiveTerm","anchor":"OeisA63880___isPrimitiveTerm","docHtml":"\n A term $n$ is primitive if no proper divisor of $n$ is in the sequence.
"},"OeisA63880.a_108":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___a_108","anchor":"OeisA63880___a_108","docHtml":"\n $108$ is in the sequence A063880.
"},"OeisA63880.a_540":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___a_540","anchor":"OeisA63880___a_540","docHtml":"\n $540$ is in the sequence A063880.
"},"OeisA63880.a_756":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___a_756","anchor":"OeisA63880___a_756","docHtml":"\n $756$ is in the sequence A063880.
"},"OeisA63880.isPrimitiveTerm_108":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___isPrimitiveTerm_108","anchor":"OeisA63880___isPrimitiveTerm_108","docHtml":"\n $108$ is a primitive term.
"},"OeisA63880.mod_216_of_a":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___mod_216_of_a","anchor":"OeisA63880___mod_216_of_a","docHtml":"\n All members of the sequence satisfy $n \\equiv 108 \\pmod{216}$.
"},"OeisA63880.powerful_of_isPrimitiveTerm":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___powerful_of_isPrimitiveTerm","anchor":"OeisA63880___powerful_of_isPrimitiveTerm","docHtml":"\n All primitive terms are powerful numbers.
"},"OeisA63880.unique_primitive_108":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___unique_primitive_108","anchor":"OeisA63880___unique_primitive_108","docHtml":"\n $108$ is the only primitive term.
"},"OeisA63880.a_of_primitive_mul_squarefree":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___a_of_primitive_mul_squarefree","anchor":"OeisA63880___a_of_primitive_mul_squarefree","docHtml":"\n If $m$ is a primitive term and $s$ is squarefree with $\\gcd(m, s) = 1$, then $m \\cdot s$\nis in the sequence.
"},"OeisA63880.exists_primitive_of_a":{"url":"/FormalConjectures/OEIS/«63880»/#OeisA63880___exists_primitive_of_a","anchor":"OeisA63880___exists_primitive_of_a","docHtml":"\n Non-primitive terms have the form $m \\cdot s$ where $m$ is primitive and $s$ is\nsquarefree with $\\gcd(m, s) = 1$.
"},"OeisA41.p":{"url":"/FormalConjectures/OEIS/«41»/#OeisA41___p","anchor":"OeisA41___p","docHtml":"\n The n-th partition number.
\n There are no partition numbers $p(k)$ of the form $x^m$, with $x,m$ integers $>1$.\nSee comment by Zhi-Wei Sun (Dec 02 2013).
"},"OeisA87719.countExceeding":{"url":"/FormalConjectures/OEIS/«87719»/#OeisA87719___countExceeding","anchor":"OeisA87719___countExceeding","docHtml":"\n Count of numbers k in {1, ..., m} where k > (minFac k)^n.
"},"OeisA87719.countNotExceeding":{"url":"/FormalConjectures/OEIS/«87719»/#OeisA87719___countNotExceeding","anchor":"OeisA87719___countNotExceeding","docHtml":"\n Count of numbers k in {1, ..., m} where k ≤ (minFac k)^n.
"},"OeisA87719.a_exists":{"url":"/FormalConjectures/OEIS/«87719»/#OeisA87719___a_exists","anchor":"OeisA87719___a_exists","docHtml":"\n There exists m such that countExceeding n m > countNotExceeding n m.
"},"OeisA87719.a":{"url":"/FormalConjectures/OEIS/«87719»/#OeisA87719___a","anchor":"OeisA87719___a","docHtml":"\n The sequence a(n): least m such that countExceeding n m > countNotExceeding n m.
"},"OeisA87719.a_one":{"url":"/FormalConjectures/OEIS/«87719»/#OeisA87719___a_one","anchor":"OeisA87719___a_one","docHtml":"\n a(1) = 15.
"},"OeisA87719.a_two":{"url":"/FormalConjectures/OEIS/«87719»/#OeisA87719___a_two","anchor":"OeisA87719___a_two","docHtml":"\n a(2) = 27.
"},"OeisA87719.a_three":{"url":"/FormalConjectures/OEIS/«87719»/#OeisA87719___a_three","anchor":"OeisA87719___a_three","docHtml":"\n a(3) = 57.
"},"OeisA87719.a_formula":{"url":"/FormalConjectures/OEIS/«87719»/#OeisA87719___a_formula","anchor":"OeisA87719___a_formula","docHtml":"\n Conjecture: a(n) = 3^n + 3 * 2^n + 6 for n ≥ 1.
"},"OeisA67720.a":{"url":"/FormalConjectures/OEIS/«67720»/#OeisA67720___a","anchor":"OeisA67720___a","docHtml":"\n A number $k$ is in the sequence A067720 if $\\varphi(k^2 + 1) = k \\cdot \\varphi(k + 1)$.
"},"OeisA67720.a_1":{"url":"/FormalConjectures/OEIS/«67720»/#OeisA67720___a_1","anchor":"OeisA67720___a_1","docHtml":"\n $1$ is in the sequence A067720.
"},"OeisA67720.a_2":{"url":"/FormalConjectures/OEIS/«67720»/#OeisA67720___a_2","anchor":"OeisA67720___a_2","docHtml":"\n $2$ is in the sequence A067720.
"},"OeisA67720.a_4":{"url":"/FormalConjectures/OEIS/«67720»/#OeisA67720___a_4","anchor":"OeisA67720___a_4","docHtml":"\n $4$ is in the sequence A067720.
"},"OeisA67720.a_6":{"url":"/FormalConjectures/OEIS/«67720»/#OeisA67720___a_6","anchor":"OeisA67720___a_6","docHtml":"\n $6$ is in the sequence A067720.
"},"OeisA67720.a_8":{"url":"/FormalConjectures/OEIS/«67720»/#OeisA67720___a_8","anchor":"OeisA67720___a_8","docHtml":"\n $8$ is in the sequence A067720.
"},"OeisA67720.a_10":{"url":"/FormalConjectures/OEIS/«67720»/#OeisA67720___a_10","anchor":"OeisA67720___a_10","docHtml":"\n $10$ is in the sequence A067720.
"},"OeisA67720.a_of_primes":{"url":"/FormalConjectures/OEIS/«67720»/#OeisA67720___a_of_primes","anchor":"OeisA67720___a_of_primes","docHtml":"\n If $k + 1$ and $k^2 + 1$ are both prime, then $k$ is in the sequence.
"},"OeisA67720.prime_add_one_of_a":{"url":"/FormalConjectures/OEIS/«67720»/#OeisA67720___prime_add_one_of_a","anchor":"OeisA67720___prime_add_one_of_a","docHtml":"\n For members of the sequence other than $8$, we have $k + 1$ is prime.
"},"OeisA281976.IsSumOfFourSquaresWithSquareConditions":{"url":"/FormalConjectures/OEIS/«281976»/#OeisA281976___IsSumOfFourSquaresWithSquareConditions","anchor":"OeisA281976___IsSumOfFourSquaresWithSquareConditions","docHtml":"\n The predicate that n can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative\nintegers, $z \\leq w$, such that both $x$ and $x + 24y$ are squares.
\nZhi-Wei Sun's Conjecture (A281976): Any integer $n \\geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$\nwith $x, y, z, w$ nonnegative integers and $z \\leq w$, such that both $x$ and $x + 24y$ are squares.
"},"OeisA81091.isPrimeBitsSet":{"url":"/FormalConjectures/OEIS/«81091»/#OeisA81091___isPrimeBitsSet","anchor":"OeisA81091___isPrimeBitsSet","docHtml":"\n Primes with $m$ one bits in their binary representation.
"},"OeisA81091.conjectureA81091":{"url":"/FormalConjectures/OEIS/«81091»/#OeisA81091___conjectureA81091","anchor":"OeisA81091___conjectureA81091","docHtml":"\nConjecture (A81091): There are infinite primes of the form $2^n + 2^i + 1$,\nwith $0 < i < n$.
"},"OeisA56777.a":{"url":"/FormalConjectures/OEIS/«56777»/#OeisA56777___a","anchor":"OeisA56777___a","docHtml":"\n A composite number $n$ is in the sequence A56777 if it satisfies both\n$\\varphi(n+12) = \\varphi(n) + 12$ and $\\sigma(n+12) = \\sigma(n) + 12$.
"},"OeisA56777.ComesFromPrimeQuadruple":{"url":"/FormalConjectures/OEIS/«56777»/#OeisA56777___ComesFromPrimeQuadruple","anchor":"OeisA56777___ComesFromPrimeQuadruple","docHtml":"\n A number $n$ comes from a prime quadruple $(p, p+2, p+6, p+8)$ if\n$n = p(p+8)$ for some prime $p$ where $p$, $p+2$, $p+6$, $p+8$ are all prime.
"},"OeisA56777.a_65":{"url":"/FormalConjectures/OEIS/«56777»/#OeisA56777___a_65","anchor":"OeisA56777___a_65","docHtml":"\n $65$ is in the sequence A56777.
"},"OeisA56777.a_209":{"url":"/FormalConjectures/OEIS/«56777»/#OeisA56777___a_209","anchor":"OeisA56777___a_209","docHtml":"\n $209$ is in the sequence A56777.
"},"OeisA56777.a_of_comesFromPrimeQuadruple":{"url":"/FormalConjectures/OEIS/«56777»/#OeisA56777___a_of_comesFromPrimeQuadruple","anchor":"OeisA56777___a_of_comesFromPrimeQuadruple","docHtml":"\n Numbers coming from prime quadruples are in the sequence A56777.
"},"OeisA56777.comesFromPrimeQuadruple_of_a":{"url":"/FormalConjectures/OEIS/«56777»/#OeisA56777___comesFromPrimeQuadruple_of_a","anchor":"OeisA56777___comesFromPrimeQuadruple_of_a","docHtml":"\n All members of the sequence A56777 come from prime quadruples.
"},"OeisA56777.mod_72_of_comesFromPrimeQuadruple":{"url":"/FormalConjectures/OEIS/«56777»/#OeisA56777___mod_72_of_comesFromPrimeQuadruple","anchor":"OeisA56777___mod_72_of_comesFromPrimeQuadruple","docHtml":"\n Numbers coming from prime quadruples satisfy $n \\equiv 65 \\pmod{72}$.
"},"OeisA56777.mod_100_of_comesFromPrimeQuadruple":{"url":"/FormalConjectures/OEIS/«56777»/#OeisA56777___mod_100_of_comesFromPrimeQuadruple","anchor":"OeisA56777___mod_100_of_comesFromPrimeQuadruple","docHtml":"\n Numbers coming from prime quadruples satisfy $n \\equiv 9 \\pmod{100}$,\nexcept the first value \"65\".
"},"OeisA231201.PrimeCondition":{"url":"/FormalConjectures/OEIS/«231201»/#OeisA231201___PrimeCondition","anchor":"OeisA231201___PrimeCondition","docHtml":"\n The predicate that n can be written as $x + y$ with $x,y >0$ such that\n$2^x + y$ is prime
\n The conjecture for sequence A231201: for any $n > 1$, there exist $x, y > 0$ such that $n = x + y$ and $2^x + y$ is prime.
"},"OeisA303656.IsSumOfTwoSquaresAndPowersOf3And5":{"url":"/FormalConjectures/OEIS/«303656»/#OeisA303656___IsSumOfTwoSquaresAndPowersOf3And5","anchor":"OeisA303656___IsSumOfTwoSquaresAndPowersOf3And5","docHtml":"\n The predicate that n can be written as $a^2 + b^2 + 3^c + 5^d$ for nonnegative integers.
\nZhi-Wei Sun's Conjecture (A303656): Any integer $n > 1$ can be written as the sum of two\nsquares, a power of 3, and a power of 5.
"},"OeisA357513.a":{"url":"/FormalConjectures/OEIS/«357513»/#OeisA357513___a","anchor":"OeisA357513___a","docHtml":"\n A357513: $a(n) = \\text{numerator of }\n\\sum_{k = 1..n} \\frac{1}{k^3} \\binom{n}{k}^2 \\binom{n+k}{k}^2\n\\text{ for } n \\ge 1 \\text{ with } a(0) = 0$.
"},"OeisA357513.zero":{"url":"/FormalConjectures/OEIS/«357513»/#OeisA357513___zero","anchor":"OeisA357513___zero"},"OeisA357513.one":{"url":"/FormalConjectures/OEIS/«357513»/#OeisA357513___one","anchor":"OeisA357513___one"},"OeisA357513.two":{"url":"/FormalConjectures/OEIS/«357513»/#OeisA357513___two","anchor":"OeisA357513___two"},"OeisA357513.three":{"url":"/FormalConjectures/OEIS/«357513»/#OeisA357513___three","anchor":"OeisA357513___three"},"OeisA357513.four":{"url":"/FormalConjectures/OEIS/«357513»/#OeisA357513___four","anchor":"OeisA357513___four"},"OeisA357513.five":{"url":"/FormalConjectures/OEIS/«357513»/#OeisA357513___five","anchor":"OeisA357513___five"},"OeisA357513.a357513_supercongruence":{"url":"/FormalConjectures/OEIS/«357513»/#OeisA357513___a357513_supercongruence","anchor":"OeisA357513___a357513_supercongruence","docHtml":"\n We have $a(p-1) \\equiv 0 \\pmod{p^4}$ for all primes $p \\ge 3$ except $p=7$.
\n\n proved by AlphaProof
"},"OeisA357513.u":{"url":"/FormalConjectures/OEIS/«357513»/#OeisA357513___u","anchor":"OeisA357513___u","docHtml":"\n Let m be a nonnegative integer and set $u(n) = $$the numerator of\n$$\\sum{k=1}^{n} \\frac{1}{k^{2m+1}} \\binom{n}{k}^2 \\binom{n+k}{k}^2$$\n(seems like a typo in the OEIS entry: the sum starts with $k=0$ there. In order\nto avoid a division by zero, we replace start the sum at $k=1$.)
"},"OeisA357513.general_supercongruence":{"url":"/FormalConjectures/OEIS/«357513»/#OeisA357513___general_supercongruence","anchor":"OeisA357513___general_supercongruence","docHtml":"\n We conjecture that $u(p-1) == 0 (mod p^4)$ for all primes $p$,\nwith a finite number of exceptions that depend on $m$.
"},"OeisA357513.general_supercongruence_one_of_a357513_supercongruence":{"url":"/FormalConjectures/OEIS/«357513»/#OeisA357513___general_supercongruence_one_of_a357513_supercongruence","anchor":"OeisA357513___general_supercongruence_one_of_a357513_supercongruence"},"OeisA232174.HasPrimeRepresentation":{"url":"/FormalConjectures/OEIS/«232174»/#OeisA232174___HasPrimeRepresentation","anchor":"OeisA232174___HasPrimeRepresentation","docHtml":"\n The predicate that n can be written as $x + y$ with $x, y > 0$ such that both\n$x + ny$ and $x^2 + ny^2$ are prime.
\nZhi-Wei Sun's Conjecture (A232174): Any integer $n > 1$ can be written as $x + y$ with\n$x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime.
"},"OeisA358684.a":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___a","anchor":"OeisA358684___a","docHtml":"\n A358684: $a(n)$ is the minimum integer $k$ such that the smallest prime factor of the $n$-th Fermat\nnumber exceeds $2^{2^n - k}$.\nLet $F_n = 2^{2^n} + 1$ be the $n$-th Fermat number, and $P_n$ be its smallest prime factor.\nThe definition of $a(n)$ is equivalent to the closed form:\n$$a(n) = 2^n - \\lfloor \\log_2(P_n) \\rfloor$$\nwhere $P_n = \\operatorname{minFac}(\\operatorname{fermatNumber} n)$.\nThe subtraction is defined in $\\mathbb{N}$ and is safe\nsince $P_n \\le F_n$, implying $\\log_2 P_n < 2^n$.
"},"OeisA358684.a'":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___a___","anchor":"OeisA358684___a___","docHtml":"\n The \"original\" definition: $a'(n)$ is the minimum $k$ such that $P_n > 2^{2^n - k}$.\nWe use Nat.find which returns the smallest natural number satisfying a predicate.
\n The log2 of the smallest prime factor of $F_n$ is at most $2^n$.
"},"OeisA358684.a_equiv_a'":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___a_equiv_a___","anchor":"OeisA358684___a_equiv_a___","docHtml":"\n The minimization definition is equivalent to the closed form.
"},"OeisA358684.zero":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___zero","anchor":"OeisA358684___zero"},"OeisA358684.one":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___one","anchor":"OeisA358684___one"},"OeisA358684.two":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___two","anchor":"OeisA358684___two"},"OeisA358684.three":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___three","anchor":"OeisA358684___three"},"OeisA358684.four":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___four","anchor":"OeisA358684___four"},"OeisA358684.five":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___five","anchor":"OeisA358684___five"},"OeisA358684.six":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___six","anchor":"OeisA358684___six"},"OeisA358684.seven":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___seven","anchor":"OeisA358684___seven"},"OeisA358684.oeis_358684_conjecture_0":{"url":"/FormalConjectures/OEIS/«358684»/#OeisA358684___oeis_358684_conjecture_0","anchor":"OeisA358684___oeis_358684_conjecture_0","docHtml":"\n Conjecture: the dyadic valuation of A93179(n) - 1 does not exceed 2^n - a(n).
\n\n A93179(n) is minFac(fermatNumber n), the smallest prime factor of the n-th Fermat number.\nThe conjecture states that if $P_n$ is the smallest prime factor of the $n$-th Fermat number,\nthen $\\nu_2(P_n - 1) \\le 2^n - a(n)$.\nSubstituting the definition of $a(n)$, this is equivalent to $\\nu_2(P_n - 1) \\le \\lfloor \\log_2(P_n) \\rfloor$.
\n\n This is Conjecture 3.4 in [SA22].
"},"OeisA308734.IsSumOfFourSquaresWithPowers":{"url":"/FormalConjectures/OEIS/«308734»/#OeisA308734___IsSumOfFourSquaresWithPowers","anchor":"OeisA308734___IsSumOfFourSquaresWithPowers","docHtml":"\n The predicate that n can be written as $(2^a \\cdot 3^b)^2 + (2^c \\cdot 5^d)^2 + x^2 + y^2$\nfor nonnegative integers $a, b, c, d, x, y$.
\nZhi-Wei Sun's Four-Square Conjecture (A308734): Any integer $n > 1$ can be written as\n$(2^a \\cdot 3^b)^2 + (2^c \\cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, x, y$.
"},"Subsets.FC100OpenSet1.problems":{"url":"/FormalConjectures/Subsets/FC100OpenSet1/#Subsets___FC100OpenSet1___problems","anchor":"Subsets___FC100OpenSet1___problems","docHtml":"\n A random subset of 100 open research problems, drawn uniformly at random\nfrom all problems with the category research open tag.
\n A random subset of 100 non-open problems, drawn uniformly at random\nfrom all problems without the category research open tag\n(solved, test, API, etc.).