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Erdős problem 401

Is there some function f(r)f(r) such that f(r)→∞f(r)\to \infty as r→∞r\to\infty, such that, for infinitely many nn, there exist a1,a2a_1,a_2 with a1+a2>n+f(r)log⁡na_1+a_2> n+f(r)\log n such that a1!a2!∣n!2n3n⋯prna_1!a_2! \mid n!2^n3^n\cdots p_r^n?

No current result

No reviewed Result is current in Vela Mathematics Program. Retained source material is shown below.

Retained declaration

FormalConjectures/ErdosProblems/401.lean

Formal Conjectures

FormalConjectures/ErdosProblems/401.leanErdos401.erdos_40112 linesExact file
True ↔  ∃ f,    Filter.Tendsto f Filter.atTop Filter.atTop ∧      ∀ (r : ℕ),        1 ≤ r →          {n |              ∃ a₁ a₂,                0 < a₁ ∧                  0 < a₂ ∧                    ↑a₁ + ↑a₂ > ↑n + f r * Real.log ↑n ∧                      a₁.factorial * a₂.factorial ∣                        n.factorial * (∏ i ∈ Finset.range r, Nat.nth Nat.Prime i) ^ n}.Infinite
SolvedProof has a holelean4external proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.

Proof manifests naming this Problem

  • Jayyhk Erdős Leanjayyhk:erdos:401
  • PLBY Lean proofsErdosProblems.Erdos401

Reported activity

Work these sources record against this Problem. Source-reported attribution, not reviewed here.

  • AI collaborating with humans

    Erdős AI contributions wiki · 10-11 Jan, 2026

    Machine
    Aristotle, GPT-5.2 Pro
    People
    Boris Alexeev, Kevin Barreto, Liam Price, Nat Sothanaphan
    Open the source record

  • argument

    VibeMathed

    Machine
    Aristotle, GPT-5.2 Pro
    People
    Boris Alexeev, Kevin Barreto, Liam Price, Nat Sothanaphan
    Reported outcome
    resolved
    Open the source record

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