FormalConjectures.ErdosProblems.«429»

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import FormalConjecturesUtil

Erdős Problem 429

References:

erdosproblems.com/429

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

[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

[We24] D. Weisenberg, Sparse Admissible Sets and a Problem of Erdős and Graham. Integers (2024).

open Filternamespace Erdos429

Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists some $n$ such that $n+a$ is prime for all $a\in A$?

Weisenberg [We24] has shown the answer is no: $A$ can be arbitrarily sparse and missing at least one residue class modulo every prime $p$, and yet $A+n$ is not contained in the primes for any $n\in \mathbb{Z}$. (Weisenberg gives several constructions of such an $A$.)

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos429.lean"]
theorem erdos_429 : answer(False) ↔
    ∃ f : ℕ → ℕ, Tendsto f atTop atTop ∧
      ∀ A : Set ℕ, A.Infinite → (∀ N, (A ∩ Set.Icc 1 N).ncard ≤ f N) →
        (∀ p : ℕ, p.Prime → ∃ b : ZMod p, ∀ a ∈ A, (a : ZMod p) ≠ b) →
        ∃ n : ℕ, ∀ a ∈ A, (n + a).Prime := by⊢ False ↔
  ∃ f,
    Tendsto f atTop atTop ∧
      ∀ (A : Set ℕ),
        A.Infinite →
          (∀ (N : ℕ), (A ∩ Set.Icc 1 N).ncard ≤ f N) →
            (∀ (p : ℕ), Nat.Prime p → ∃ b, ∀ a ∈ A, ↑a ≠ b) → ∃ n, ∀ a ∈ A, Nat.Prime (n + a)
  sorryAll goals completed! 🐙

end Erdos429