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We show that for some $k\le 3570$ and all $k$ with $442720643463713815200|k$,the equation $\phi(n)=\phi(n+k)$ has infinitely many solutions $n$, where$\phi$ is Euler's totient function. We also show that for a positive proportionof 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 byZhang, Maynard, Tao and PolyMath.

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