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Erdos Problem #1061

Number theory · posed by Paul Erdos · disproved

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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.

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

  1. proof attempt · #1

    Eric Li, using ChatGPT, Aristotle

    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.

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    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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