  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.
