A resolution of Erdős Problem 1061 on the sum-of-divisors function

We resolve Erdős Problem 1061, the question whether the number $$S(x)=\mathrm{\#}\{(a,b)\in {\mathbb{N}}^2:a+b\le x,\sigma (a)+\sigma (b)=\sigma (a+b)\}$$ of ordered solutions has a linear asymptotic $S(x)\sim cx$. In fact the opposite extreme holds at every fixed logarithmic scale: for every $R>0$, $$\underset{x\to \mathrm{\infty }}{\lim }\frac{S(x)}{x(\log x{)}^R}=+\mathrm{\infty }\mathrm{.}$$ 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.