Problem 830

We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$ . Are there infinitely many amicable pairs? If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then is it true that

A(x)>x^{1-o(1)}?

Source: erdosproblems.com/830

Number theory

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Er55b: Erdős, P., On amicable numbers. Publ. Math. Debrecen (1955), 108-111.
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 B4 "Amicable numbers" is on pp. 86--87; the conjecture $A(x)\geq x^{1-\epsilon}$ and the bounds $A(x)=o(x)$ and $A(x)\ll x\exp\{-(\ln x)^{1/3}\}$ are on p. 87. Library home: Guy 2004.
Po15: Pomerance, Carl, On amicable numbers. Analytic Number Theory, Springer (2015), 321-327; doi:10.1007/978-3-319-22240-0_19.
Po81: Pomerance, Carl, On the distribution of amicable numbers. II. J. Reine Angew. Math. 325 (1981), 183-188; doi:10.1515/crll.1981.325.183.

Problem 830

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