The minimal number of solutions to $\phi (n)=\phi (n+k)$

Abstract:

In 1958, A. Schinzel showed that for each fixed $k\leq 8\cdot 10^{47}$ there are at least two solutions to $\phi (n)=\phi (n+k)$. Using the same method and a computer search, Schinzel and A. Wakulicz extended the bound to all $k \leq 2\cdot 10^{58}$. Here we show that Schinzel’s method can be used to further extend the bound when $k$ is even, but not when $k$ is odd.