Mathematics

We fix a positive integer $k$ and look for solutions $n \in \mathbb{N}$ of the equations $\phi(n + k) = \phi(n)$ and $φ(n + k) = 2 φ(n)$. For $k \le 12 \cdot 10^{100}$, 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. Furthermore, for $k \le 4 \cdot 10^{58}$, we show that for the second equation there are at least three solutions when $k$ is even. Our work increases the previously known minimal number of solutions for both equations.