OPEN
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
\[A(x)>x^{1-o(1)}?\]
For example $220$ and $284$. Erdős
[Er55b] proved that $A(x)=o(x)$, and Pomerance
[Po81] improved this to
\[A(x) \leq x \exp(-(\log x)^{1/3})\]
and later
[Po15] to
\[A(x) \leq x \exp(-(\tfrac{1}{2}+o(1))(\log x\log\log x)^{1/2}).\]
