Abstract: In this paper we investigate the solutions of the equation in the title, where $\phi $ is the Euler function. We first show that it suffices to find the solutions of the above equation when $m=4$ and $x$ and $y$ are coprime positive integers. For this last equation, we show that aside from a few small solutions, all the others are in a one-to-one correspondence with the Fermat primes.
Keywords: Euler function, Fermat primes
Classification (MSC2000): 11A25, 11A51, 11A63
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