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Erdős problem 728

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Whether there are infinitely many integers a,b,na, b, n with a,b≥εna, b \ge \varepsilon n such that a!⋅b!a!\cdot b! divides n!⋅(a+b−n)!n!\cdot(a+b-n)! while a+ba+b exceeds nn by more than C⋅log⁡nC\cdot\log n.

No current result

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

Retained declaration

FormalConjectures/ErdosProblems/728.lean

Formal Conjectures

FormalConjectures/ErdosProblems/728.leanErdos728.erdos_72810 linesExact file
True ↔  ∀ᶠ (ε : ℝ) in nhdsWithin 0 (Set.Ioi 0),    ∀ C > 0,      ∀ C' > C,        ∃ a b n,          0 < n ∧            ε * ↑n < ↑a ∧              ε * ↑n < ↑b ∧                a.factorial * b.factorial ∣ n.factorial * (a + b - n).factorial ∧                  ↑a + ↑b > ↑n + C * Real.log ↑n ∧ ↑a + ↑b < ↑n + C' * Real.log ↑n
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:728
  • PLBY Lean proofsErdosProblems.Erdos728
  • PLBY Lean proofsErdosProblems.Erdos728p

Reported activity

Reported by these sources. Not reviewed here.

  • AI alongside literature

    Erdős AI contributions wiki · 6 Jan, 2026

    Machine
    Aristotle, GPT-5.2 Pro
    Open the source record

  • Formalization

    Erdős AI contributions wiki · 22 Jan, 2026

    Machine
    Aristotle
    Open the source record

  • argument

    VibeMathed

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

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