Problem #48: Are there infinitely many integers $n,m$ such that...

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Progress

This would follow immediately from the twin prime conjecture. The answer is yes, proved by Ford, Luca, and Pomerance [FLP10], who in fact prove there are at least\[\exp((\log\log x)^c)\]many $a\leq x$ such that $a=\phi(n)=\sigma(m)$ for some $n,m$, where $c>0$ is an absolute constant. This lower bound was improved to\[\exp((\log\log x)^{\omega(x)})\]for some $\omega(x)\to \infty$ by Garaev [Ga11].

This is problem B38 of Guy's collection [Gu04].