Problem 1061

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claims/: The 1 claim page of Problem 1061, one per claimant's result; the problem's standing derives from them.

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Statement. How many solutions are there to

$$\sigma(a)+\sigma(b)=\sigma(a+b)$$

with $a+b\leq x$, where $\sigma$ is the sum of divisors function? Is it $\sim cx$ for some constant $c>0$?

Statement (precise). How many solutions are there to

$$\sigma(a)+\sigma(b)=\sigma(a+b)$$

with $a+b\leq x$, where $\sigma$ is the sum of divisors function? Is it $\sim cx$ for some constant $c>0$, or of higher order?

Notes. The site's first question is ambiguous between asking for the order of growth and asking Guy's dichotomy. Guy [Gu04], B15, p. 105, fixes it as the dichotomy, reporting Erdős's question as whether the number of solutions with $a+b\le x$ is $cx+o(x)$ or of higher order, and the formal-conjectures statement erdos_1061 formalizes the same yes-or-no question, whether $S(x)\sim cx$ for some $c>0$. Ordered pairs are counted. There are no solutions with $a=b$ (Remark 1.2 of Li 2026), so the unordered count is exactly half. A proof that the count exceeds $x(\log x)^R$ for every $R>0$ answers the precise Statement in the higher-order direction, and the answer to whether the count is $\sim cx$ is no. Such a bound does not determine the order of growth, which remains open.

Status. Open. Li's preprint of June 2026 (arXiv:2606.25849, written with GPT-5.5 Pro) claims that the number of solutions with $a+b\le x$ exceeds $x(\log x)^{R}$ for every fixed $R>0$, so that no asymptotic $cx$ holds; the author posted it as a full proof claim on the site's proof-claims tab on 2026-07-26. The site labels the problem OPEN; no outside review is known, and the claim is pending on Li 2026 (proof-claims thread accessed 2026-10-06).

Source. erdosproblems.com/1061, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1061, https://www.erdosproblems.com/1061.

References.

• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section B15 "Solutions of $\sigma(q)+\sigma(r)=\sigma(q+r)$", p. 105, asks how many solutions there are with $q+r<x$, whether $cx+o(x)$ or of higher order. Library home: guy_2004_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures (erdos_1061, tagged research open and proved by sorry at the pinned commit; no formal proof is filed there).

Current assessment

The question (site formulation). How many pairs $(a,b)$ with $a+b\le x$ satisfy $\sigma(a)+\sigma(b)=\sigma(a+b)$, and whether the count is $\sim cx$ for some constant $c>0$. The site labels the problem OPEN.

Standing. Claimed, disproved: the full claim Li 2026 (arXiv:2606.25849, first version submitted 2026-06-24, written with GPT-5.5 Pro) asserts that the count exceeds $x(\log x)^{R}$ for every fixed $R>0$, so that no asymptotic $cx$ holds. The author posted it as a full proof claim on the site's proof-claims tab on 2026-07-26; the thread carried no comments as of 2026-10-06, and no referee report or outside review of the proof is known. The claim's full scope rests on the precise Statement above: it answers Guy's dichotomy and the question whether the count is $\sim cx$, but it does not determine the order of growth.

Lean coverage. The formal-conjectures statement erdos_1061, at the commit pinned in the Formalization paragraph, states the question with answer(sorry) and a sorry proof, and no formal proof is filed there; no file in the lean-proofs repository treats the problem.

Historical record. Guy [Gu04], section B15, records the question as how many solutions there are with $q+r<x$, whether $cx+o(x)$ or of higher order; in that formulation the Li claim asserts the count is of higher order.

Search scope. The site's problem page, accessed 2026-09-04, and its proof-claims thread, accessed 2026-10-06; the formal-conjectures file at the pinned commit and the lean-proofs repository, as of 2026-10-07; and Guy's B15. The proof is not compiled in this wiki.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• li_2026_resolution_erdos_problem_1061_sum_divisors
• li_2026_resolution_erdos_problem_1061_sum_divisors / remark_1_2
• li_2026_resolution_erdos_problem_1061_sum_divisors / theorem_1_1
• li_2026_resolution_erdos_problem_1061_sum_divisors / theorem_1_3
• guy_2004_unsolved_problems_number_theory