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Erdős Problem #26

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erdos-26Number theoryposed by Paul Erdős, Gérald Tenenbaum, 1995recorded: variant

1 attempt · 1 machine check · no person has looked

Statement

Let A⊂NA\subset\mathbb{N} be infinite. Must there exist some k≥1k\geq 1 such that almost all integers have a divisor of the form a+ka+k for some a∈Aa\in A? The question as posed follows negatively from Davenport–Erdős (1951). The AI result settles Tenenbaum's harder variant, also negatively: there is an infinite AA such that for every k≥1k\geq 1 the set of multiples of A+kA+k has upper density below 0.340.34.

Context

The question as posed was implicit in Davenport–Erdős (1951); the AI result settles Tenenbaum's open variant negatively

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.

People

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Projects

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Interest

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Attempts

1 attempt

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  • #1

    Attempt 1

    constructionDeepMind prover agent ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    DeepMind prover agent

    A DeepMind prover agent constructed an infinite set AA such that for every k≥1k\geq 1 the set of multiples of A+kA+k has upper density less than 0.340.34, resolving Tenenbaum's variant of the problem in the negative.

    The question as posed was implicit in Davenport–Erdős (1951); the AI result settles Tenenbaum's open variant negatively

    Reviews

    0 human reviews · 1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: correct

      Recorded from VibeMathed site check ·

      scope Reproduction by the VibeMathed site

      erdosproblems.com marks the problem DISPROVED and documents the DeepMind construction in the page remarks; the variant result is recorded there without a separate formal artifact.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

    Endorsements

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Discussion

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