Solutions of $\varphi (n)=\varphi (n+k)$ and $\sigma (n)=\sigma (n+k)$

We show that for some $k\le 3570$ and all $k$ with $442720643463713815200\mathrm{\mid }k$, the equation $\varphi (n)=\varphi (n+k)$ has infinitely many solutions $n$, where $\varphi$ is Euler’s totient function. We also show that for a positive proportion of all $k$, the equation $\sigma (n)=\sigma (n+k)$ has infinitely many solutions $n$. The proofs rely on recent progress on the prime $k$-tuples conjecture by Zhang, Maynard, Tao and PolyMath.