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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 primescan be used to build five solutions for the first equation when $k$ is even andfive for the second one when $k$ is odd. These results hold for $k \le 2 \cdot10^{100}$. We also show that for the second equation with even $k$ there are atleast three solutions for $k \le 4 \cdot 10^{58}$. Our work increases theprevious minimal number of known solutions for both equations.

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