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

TL;DR

This paper investigates solutions to the equations involving Euler's totient function for fixed shifts, establishing new minimal solution counts and linking solutions to Fermat primes for large ranges of k.

Contribution

It demonstrates that Fermat primes can generate multiple solutions for the equations and extends the known minimal number of solutions for large values of k.

Findings

Fermat primes enable construction of five solutions for even k in the first equation.

Five solutions are possible for odd k in the second equation using Fermat primes.

At least three solutions exist for the second equation with even k up to 4×10^{58}.

Abstract

We fix a positive integer $k$ and look for solutions of the equations $\phi(n+k) = \phi(n)$ and $\phi(n + k) = 2\phi(n)$. 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.