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Erdős Problem #728: Factorial Divisibility

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

1 attempt · 1 machine check

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.

Context

A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.

People

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Attempts

1 attempt

No person has examined this. There is 1 attempt here and 1 machine check recorded against it. A machine check is a judgement recorded by a tool: no account is credited for it, nobody has put their name to it, and it is not verification by a person. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

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  • #1

    Attempt 1

    proof attemptAristotle (Harmonic) + GPT-5.2 Pro with Boris Alexeev, Kevin Barreto, Liam Price, Nat Sothanaphan ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    Aristotle (Harmonic) + GPT-5.2 Pro
    people
    Boris Alexeev, Kevin Barreto, Liam Price, Nat Sothanaphan

    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.

    Reviews

    1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: correct

      Recorded from Lean ·

      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.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

    Discussion of this attempt

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