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Erdős Problem #793

Number theory · posed by Paul Erdős, 1969 · solved

1 attempt · 1 machine check

Statement

Let F(n)F(n) be the largest A⊆{1,…,n}A\subseteq\{1,\dots,n\} with a∤bca\nmid bc for distinct a,b,c∈Aa,b,c\in A. Is F(n)=π(n)+(C+o(1)) n2/3(log⁡n)−2F(n)=\pi(n)+(C+o(1))\,n^{2/3}(\log n)^{-2} for some constant CC?

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 attemptGPT-5.6 Sol with Przemek Chojecki ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.6 Sol
    people
    Przemek Chojecki

    GPT-5.6 Sol (prompted by Przemek Chojecki) proved F(n)=π(n)+(272+o(1))n2/3(log⁡n)2F(n)=\pi(n)+(\tfrac{27}{2}+o(1))\frac{n^{2/3}}{(\log n)^2}, a refined form of Erdős's 1938 argument.

    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 VibeMathed site check ·

      scope Reproduction by the VibeMathed site

      Marked proved on erdosproblems.com via a proof claim by GPT-5.6 Sol; not formally Lean-verified.

      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.

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