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

Kevin Ford

Published 2020-02-27Version 1

We show that for some $k\le 6990$ and all $k$ with $3099044504245996706400|k$, 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 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.