Infinitude of amicable numbers

Prove that infinitely many amicable pairs exist, equivalently establish that the counting function A(x) for amicable numbers satisfies A(x)\to\infty as x\to\infty.

Background

The paper defines an amicable pair as two distinct positive integers n and n' satisfying s(n)=n' and s(n')=n, where s(n) is the sum of the proper divisors of n. Although more than a billion amicable pairs have been identified computationally, the existence of infinitely many such pairs has not been established.

The unresolved issue concerns the lower-bound behavior of A(x), the number of amicable numbers not exceeding x. The paper proves a strong upper bound for A(x), but explicitly notes that even the much weaker assertion that A(x) tends to infinity remains unproved.