*preprint*

**Inserted:** 26 oct 2023

**Last Updated:** 26 oct 2023

**Year:** 2021

**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.