Solutions of $ϕ(n)=ϕ(n+k)$ and $σ(n)=σ(n+k)$

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

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