A published paper or arXiv preprint answers the main question of Erdős Problem #824 (theorem erdos_824 in formal-conjectures) in full (proof, disproof, or the exact value asked for) with no hypotheses beyond the problem's own.

The problem, as stated in formal-conjectures (Apache 2.0):

Let h(x) count the number of integers 1\leq a<b<x such that (a,b)=1 and \sigma(a)=\sigma(b), where \sigma is the sum of divisors function. Is it true that h(x)>x^{2-o(1)}? That is, is it true that for every \epsilon > 0 we have h(x) > x^{2-\epsilon} for all large x? According to erdosproblems.com, Erdős [Er74b] proved that \limsup h(x)/x = \infty and claimed a similar proof for this problem.

References formal-conjectures lists for this problem (a starting point, not evidence):

• [Er59c] Erdős, P., *Remarks on number theory. II. Some problems on the \sigma function*. Acta Arith. (1959), 171--177.
• [Er74b] Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.
• [PoPo16] Pollack, Paul and Pomerance, Carl, Some problems of Erdős on the sum-of-divisors function. Trans. Amer. Math. Soc. Ser. B (2016), 1-26.