Let & denote the phi function
Show that, for any given m, the equation &(n) = m has finitely many solutions n (or no solutions)
It follows from this post that there's a finite number of positive integers $\displaystyle n$ such that $\displaystyle \phi(n)\leq m$ (1) (since $\displaystyle \phi(n)\to +\infty$ as $\displaystyle n\to +\infty$ ), hence your equation must have a finite number of solutions -since it's a subset of the integers that satisfy (1) -.