Finite fibres of the Euler totient function
= Finite fibres of the Euler totient function
For each positive integer $k$, only finitely many $n$ satisfy $\varphi(n)=k$. Indeed, every prime $p\mid n$ has $p-1\mid k$, and the exponent of $p$ in $n$ is bounded because $p^{a-1}\mid k$.