FormalConjectures.ErdosProblems.«424»

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module

public import FormalConjecturesUtil

Erdős Problem 424: Sequence generated by $a_i a_j - 1$

References:

erdosproblems.com/424

A5244

[Ben Green's Open Problem 63](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 63)

[Ko26] Korsky, S., A problem of Erdős on the sequence $a_ia_j - 1$, arXiv:2608.07910

@[expose] public sectionnamespace Erdos424open Set

Defines the set of new numbers generated from a set A by the operation $x y - 1$ for $x \neq y$.

def nextGeneration (A : Set ℕ) : Set ℕ :=
  { z : ℕ | ∃ x y, x ∈ A ∧ y ∈ A ∧ x ≠ y ∧ z = x * y - 1 }

The sequence of sets $A_n$ where $A_0 = {2, 3}$ and $A_{n+1}$ is $A_n$ union all newly generated elements.

def sequenceSet : ℕ → Set ℕ
  | 0 => {2, 3}
  | n + 1 => (sequenceSet n) ∪ (nextGeneration (sequenceSet n))

The set of integers which eventually appear in the sequence, which is the union of all $A_n$.

def generatedSet : Set ℕ := ⋃ n : ℕ, sequenceSet n

Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \ldots, a_n$ all possible values of $a_i a_j - 1$ with $i \neq j$. Is it true that the set of integers which eventually appear has positive density?

As explained on erdosproblems.com/424, "positive density" here means positive lower density: is there $c > 0$ such that for all large $x$ at least $cx$ of the integers in $[1, x]$ appear? See erdos_424.variants.exact_density for the literal reading.

Korsky [Ko26] has announced a proof that the set has positive lower density; it is listed as a proof claim on erdosproblems.com/424, which still records the problem as open.

@[category research open, AMS 11]
theorem erdos_424 : answer(sorry) ↔ 0 < generatedSet.lowerDensity := by⊢ True ↔ 0 < generatedSet.lowerDensity
  sorryAll goals completed! 🐙

A literal interpretation of "positive density": the natural density of generatedSet exists (i.e. the lower and upper density agree) and is positive.

@[category research open, AMS 11]
theorem erdos_424.variants.exact_density : answer(sorry) ↔ generatedSet.HasPosDensity := by⊢ True ↔ generatedSet.HasPosDensity
  sorryAll goals completed! 🐙

-- TODO(firsching): formalize the statements from the additional material

end Erdos424