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Erdős Problem #320

Number theory · posed by Paul Erdős, Ronald Graham, 1980 · solved

1 attempt · 1 machine check

Statement

Let S(N)S(N) count the distinct values of ∑n∈A1/n\sum_{n\in A} 1/n over A⊆{1,…,N}A\subseteq\{1,\dots,N\}. Estimate S(N)S(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 attemptGPT-5.6 Sol with Young, Zhu, Luo ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.6 Sol
    people
    Young, Zhu, Luo

    GPT-5.6 Sol (prompted by Young, Zhu, and Luo) proved a matching upper bound, pinning log⁡S(N)\log S(N) to order Nlog⁡N∏j≥3log⁡jN\frac{N}{\log N}\prod_{j\ge 3}\log_j N.

    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 solved on erdosproblems.com via a proof claim; 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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