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.