On the minimal number of solutions of the equation $\varphi (n+k)=M\varphi (n)$, $M=1$, $2$

Abstract

We fix a positive integer $k$ and look for solutions of the equations $\varphi (n+k)=\varphi (n)$ and $\varphi (n+k)=2\varphi (n)$. We prove that Fermat primes can be used to build five solutions for the first equation when $k$ is even and five for the second one when $k$ is odd. These results hold for $k\le 2\cdot 10^{100}$. We also show that for the second equation with even $k$ there are at least three solutions for $k\le 4\cdot 10^{58}$. Our work increases the previous minimal number of known solutions for both equations.

arXiv comment9 pages, no figures