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Erdős Problem #26

Number theory · posed by Paul Erdős, Gérald Tenenbaum, 1995 · variant

2 comments

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.

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

  1. construction · #1

    DeepMind prover agent

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

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

    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

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed 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.

    Repeated from the source; nothing was checked here.

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