In this paper we investigate the solutions of the equation in the title, where φ 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.
For a positive integer $n$ we write $\phi (n)$ for the Euler function of $n$. In this note, we show that if $b>1$ is a fixed positive integer, then the equation \[ \phi \Big (x\frac{b^n-1}{b-1}\Big )=y\frac{b^m-1}{b-1},\qquad {\text{where}} \ x,~y\in \lbrace 1,\ldots ,b-1\rbrace , \] has only finitely many positive integer solutions $(x,y,m,n)$.