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Erdős Problem #1201

Number theory · posed by Paul Erdős, 1976 · partial

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Statement

Is it true that for every ϵ,η>0\epsilon,\eta>0 there exists a kk such that the density of nn for which P(n(n+1)⋯(n+k))>n1−ϵP(n(n+1)\cdots(n+k))>n^{1-\epsilon} is at least 1−η1-\eta, where P(m)P(m) is the greatest prime divisor of mm? A short argument via the Matomäki-Radziwiłł theorem establishes the lower-density version.

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

  1. proof attempt · #1

    Przemysław Chojecki, using GPT-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 deduction from the Matomäki-Radziwiłł theorem on multiplicative functions was written by GPT-5.5 Pro; Tao and Sawin's forum discussion pinned down exactly what the known results do and do not give for this problem.

    As Tao notes on the problem page, the claim establishes natural LOWER density at least 1-eta but not that the natural density exists, so the problem as stated remains technically open

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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.