1. FormalConjectures
  2. ErdosProblems
  3. «473»
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Erdős Problem 473

References:

    erdosproblems.com/473

    [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique (1980).

@[expose] public sectionnamespace Erdos473

Is there a permutation $a_1, a_2, \ldots$ of the positive integers such that $a_k + a_{k+1}$ is always prime?

A question of Segal [ErGr80, p.94]. The answer is yes, as shown by Odlyzko. The linked formal proof (Codex and GPT-5.6 Sol) builds the permutation as a spanning one-way ray of the graph on the positive integers in which two numbers are adjacent when their sum is prime.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/8822f7ddef30fadbd92e1c6ab4ed897af356af5e/src/latest/ErdosProblems/Erdos473.lean#L236"] theorem erdos_473 : answer(True) ↔ ∃ a : ℕ ≃ ℕ+, ∀ n : ℕ, ((a n : ℕ) + (a (n + 1) : ℕ)).Prime := ⊢ True ↔ ∃ a, ∀ (n : ℕ), Nat.Prime (↑(a n) + ↑(a (n + 1))) All goals completed! 🐙

Segal also asked whether for every $n \ge 2$ there is a permutation $a_1, \ldots, a_n$ of ${1, \ldots, n}$ such that $a_k + a_{k+1}$ is prime for all $1 \le k < n$. This is conjectured to be true, and has been verified for infinitely many $n$.

@[category research open, AMS 11] theorem erdos_473.variants.finite : answer(sorry) ↔ ∀ n : ℕ, 2 ≤ n → ∃ a : Fin n ≃ Fin n, ∀ k : Fin n, ∀ h : k.val + 1 < n, ((a k).val + 1 + ((a ⟨k.val + 1, h⟩).val + 1)).Prime := ⊢ True ↔ ∀ (n : ℕ), 2 ≤ n → ∃ a, ∀ (k : Fin n) (h : ↑k + 1 < n), Nat.Prime (↑(a k) + 1 + (↑(a ⟨↑k + 1, h⟩) + 1)) All goals completed! 🐙end Erdos473