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Erdős Problem #387

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erdos-387-binomial-divisorsNumber theoryposed by Paul Erdős, Ronald Grahamrecorded: disproved

1 attempt · no person has looked

Statement

Erdős and Graham asked whether (nk)\binom{n}{k} with 1≤k≤n/21 \le k \le n/2 must always have a divisor ≤n\le n that is close to nn, meaning bigger than a fixed constant times nn. Settled in both directions: true when kk is large enough as a function of nn, but false in general, since there are (nk)\binom{n}{k} with kk small compared to nn having no such divisor.

Context

false in general; the positive direction holds for k large relative to n

A fifty-year-old Erdős-Graham conjecture, tracked as #387 on the Erdős problems site, resolved in both directions.

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1 attempt

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  • #1

    Attempt 1

    proof attemptChatGPT 5.5 Pro with Hung M. Bui, Slava Naprienko, Kyle Pratt, Alexandru Zaharescu ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    ChatGPT 5.5 Pro
    people
    Hung M. Bui, Slava Naprienko, Kyle Pratt, Alexandru Zaharescu

    The disclosure is carefully scoped rather than blanket. The main ideas in the proof of Theorem 5.1 were developed in interactive sessions between the authors and ChatGPT 5.5 Pro, and some documents and code in the accompanying repository were generated with AI assistance. The authors separately used ChatGPT for literature searches and for spotting typos, and they state that all text in the paper is human-generated. The heavier half of the paper, a restricted covering problem attacked with sieve methods and exponential sum estimates, is presented as the authors' own.

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