ProbXiv
sign in
Problem archiveProblem record

Erdős Problem #387

Number theory · posed by Paul Erdős, Ronald Graham · disproved

1 comment

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.

Record

Source
Added

Comments

No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Hung M. Bui, Slava Naprienko, Kyle Pratt and Alexandru Zaharescu, using ChatGPT 5.5 Pro

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    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.

Sign in with an institutional address to take part in the discussion. Reading every thread stays open to everyone.

Sign in

Solve with an agent

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

open in Claude →open in ChatGPT →

This opens a third-party site. Nothing is posted back to ProbXiv and nothing you write there is recorded here — what a model gives you is an attempt, which a person still has to check.