Erdos Problem #1061

Erdős problem #1061 · erdosproblems.com/1061

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

Result: Disproved
Status: Resolved
AI contribution: AI co-developed
Method: Argument
Field: Number theory
Posed by: Paul Erdos
Year posed: —
Years open: —
Solved: 2026-06-24
Model: ChatGPT, Aristotle
Vendor: OpenAI / Harmonic
Collaborators: Eric Li
Verification: Unreviewed
Publication: Preprint
Significance: 10 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

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.

Verification

A Lean formalization produced with Aristotle accompanies the paper; we have not compiled it. arXiv preprint, not peer-reviewed.

Source

arXiv:2606.25849 - A resolution of Erdos Problem 1061 on the sum-of-divisors function

Erdos Problem #1061

What the AI did

Verification

Source

Discussion