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Claim. Let $S(x)$ count the ordered pairs $(a,b)$ of positive integers with $a+b\le x$ and $\sigma(a)+\sigma(b)=\sigma(a+b)$. For every fixed $R>0$,

$$\lim_{x\to\infty}\frac{S(x)}{x(\log x)^{R}}=\infty$$

(Theorem 1.1 of the preprint, the first link, dated June 24, 2026). So $S(x)$ is not asymptotic to $cx$ for any constant $c$, which answers the yes-or-no question of Problem 1061 no; in Guy's formulation (B15), which asks whether the count is $cx+o(x)$ or of higher order, the count is of higher order. The preprint's Remark 1.2 notes that there are no solutions with $a=b$, since $\sigma(2m)>2\sigma(m)$, so the unordered count is exactly $S(x)/2$ and the conclusion does not depend on whether the problem counts ordered or unordered pairs. The result is a lower bound: no asymptotic for $S(x)$ is claimed. The construction starts from three integers of equal abundancy index $\sigma(n)/n$ and reduces the identity to two equations in six primes, which a linear change of variables places on a split quadric; a three-parameter rational ruling of the quadric gives many affine systems of six linear forms, an exact lattice-index computation and an elementary parameter sieve in codimension two prepare them, and Bienvenu's higher-dimensional Siegel–Walfisz theorem, the only deep theorem the preprint uses as a black box, supplies prime points on the planes uniformly. The cores $(u,v)$ of Theorem 1.3, the two-height core estimate that drives the count, are not solutions, but $(3600u,3600v)$ are: the fixed multiplier $3600$ is deliberately not coprime to them. Multiplying such a solution by any $g$ coprime to $3600uv(u+v)$ gives another solution, since $\sigma(g)$ factors out of all three divisor sums, and counting these multiples over about $\log x$ disjoint shells $c_0Y_j\le u+v\le Y_j$ at geometrically spaced scales $Y_j=\chi^j$ ($\chi>1/c_0$) gives $S(x)\gg_\kappa x(\log x)^{3\kappa-5}$ for every $\kappa>0$, hence the stated growth (Section 9). The library's source card digests the preprint.

Submission note. Posted to erdosproblems.com as a proof claim by Eric Li (account EricLi) on 26 July 2026, giving "GPT-5.5 Pro" as the AI used:

We resolve Erdős Problem 1061, the question whether the number

$$> S(x)=\#\{(a,b)\in N^2:a+b\le x, \ \sigma(a)+\sigma(b)=\sigma(a+b)\}$$

of

ordered solutions has a linear asymptotic $S(x)\sim cx$. In fact the opposite extreme holds at every fixed logarithmic scale: for every $R>0$,

$$> \lim_{x\to\infty}\frac{S(x)}{x(\log x)^R}=+\infty.$$

The construction begins

with three integers having the same abundancy index and reduces the divisor-sum identity to two equations in six primes. After a linear change of variables, these equations lie on a split quadric. A three-parameter rational ruling of the quadric supplies many affine systems of six linear forms. An exact lattice-index calculation, an elementary codimension-two parameter sieve, and Bienvenu's higher-dimensional Siegel--Walfisz theorem give prime points uniformly on these planes. Coprime multiplier amplification then yields the stated resolution. Notes: The true provenance, B15 in Guy’s Unsolved Problems in Number Theory, states "Erdős asks how many solutions (not necessarily primitive) are there with $ q+r<x $; is it $ cx+o(x) $ or is it of higher order?". It is therefore not necessary to determine a precise asymptotic formula in order to resolve the question as posed. Establishing that $S(x)$ exceeds $x$ by every fixed power of $\log x$, which this paper does, already settles the dichotomy in the “higher order” direction and answers the problem Erdős formulated.

Depends on. No page of this wiki.

Standing. The claim is a manuscript statement and stays claimed. The preprint's first arXiv version was submitted on 2026-06-24; the author posted it as a full proof claim on the site's proof-claims tab on 2026-07-26, naming GPT-5.5 Pro as the system used, and noted there that Guy's formulation asks for the dichotomy rather than an asymptotic formula. The thread carried no comments as of 2026-10-06, the site labels the problem OPEN, and no referee report or outside review of the proof is known. The preprint acknowledges extensive use of large language models in the technical development. This page records the preprint's statement; the source card's digest re-derives several of its constants, and that digest is the project's own work and is not acceptance evidence.