Problem Statement
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\[A(x)>x^{1-o(1)}?\]
Categories:
Number Theory
Progress
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\[A(x) \leq x \exp(-(\tfrac{1}{2}+o(1))(\log x\log\log x)^{1/2}).\]This is problem B4 in Guy's collection [Gu04].Source: erdosproblems.com/830 | Last verified: January 16, 2026