We fix a positive integer k and look for solutions n ∈ N of the equations φ(n + k) = φ(n) and φ(n + k) = 2φ(n). For k ≤ 12 · 10¹⁰⁰, 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 ≤ 4 · 10⁵⁸, 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.