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Erdős Problem #1201

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erdos-1201Number theoryposed by Paul Erdős, 1976recorded: partial

1 attempt · no person has looked

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.

Context

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

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.

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Attempts

1 attempt

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  • #1

    Attempt 1

    proof attemptGPT-5.5 Pro with Przemysław Chojecki ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.5 Pro
    people
    Przemysław Chojecki

    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

    Reviews

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    Discussion of this attempt

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Discussion

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