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Erdos's Conjecture on Consecutive Integers Free of Certain Prime Factors

Number theory · posed by Paul Erdos · solved

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Statement

Let nkn_k be the least n>2kn > 2k such that (n−k)(n−k+1)⋯(n−1)(n-k)(n-k+1)\cdots(n-1) has no prime factor in (k,2k)(k, 2k). Erdos conjectured a superpolynomial lower bound; for all large kk, nk>elog⁡2k/(20log⁡log⁡k)n_k > e^{\log^2 k / (20 \log\log k)}.

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

  1. proof attempt · #1

    ChatGPT 5.5 Pro, Aristotle, with Wouter van Doorn and Quanyu Tang

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    The paper says the bound was conceived of by ChatGPT 5.5 Pro and that the original paper the model wrote remains publicly available, so the provenance is checkable rather than asserted. The supporting proofs were obtained by Harmonic's Aristotle and are stated to be fully self-contained.

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