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Erdős Problem #728: Factorial Divisibility

Number theory · posed by Paul Erdős, Ronald Graham, Imre Ruzsa, Ernst Straus, 1975 · solved

2 comments

Statement

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.

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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. proof attempt · #1

    Aristotle (Harmonic) + GPT-5.2 Pro, with Boris Alexeev, Kevin Barreto, Liam Price and Nat Sothanaphan

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    Aristotle (Harmonic's Lean-based prover) and GPT-5.2 Pro produced a fully autonomous, Lean-verified proof. Regarded by the erdosproblems.com maintainers as the first Erdős problem resolved autonomously by AI systems, about three months before the more widely covered #1196 result.

  2. Machine-checked by Lean on #1 · not a person

    lean: correctLean

    scope Lean formalization of the result

    Machine-checked end to end in the Lean proof assistant - every logical step formally verified, not just human-reviewed. Written up formally by Nat Sothanaphan.

    Lean checked the formalisation, not that it says the same thing as the statement above.

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Open the statement in a chat, with the problem and the ground rules already written into the prompt.

open in Claude →open in ChatGPT →

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