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Erdos Problem #1061

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erdos-1061-sum-of-divisorsNumber theoryposed by Paul Erdosrecorded: disproved

1 attempt · no person has looked

Statement

Let S(x)S(x) count ordered pairs (a,b)(a,b) with a+b≤xa+b \le x and σ(a)+σ(b)=σ(a+b)\sigma(a)+\sigma(b) = \sigma(a+b). Erdos asked whether S(x)∼cxS(x) \sim cx. The opposite extreme holds: for every R>0R > 0, S(x)/(x(log⁡x)R)→∞S(x)/(x(\log x)^R) \to \infty, so the count beats every fixed logarithmic scale.

Context

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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Attempts

1 attempt

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  • #1

    Attempt 1

    proof attemptChatGPT, Aristotle with Eric Li ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    ChatGPT, Aristotle
    people
    Eric Li

    The declaration says large language models, primarily ChatGPT, were used extensively throughout the research, with the author originating the ideas, and that the accompanying Lean formalization was produced with Harmonic's Aristotle under the author's direction and audit.

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