Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 11I / d / Solution

By the primitive-root modulus classification, $P(n)>0$ only forwith $p$ an odd prime, and thenFix $k\ge 1$. The finite fibres of the Euler totient function imply that there are only finitely many possible values $m=\phi (n)$ satisfying $\phi (m)=k$. For each such $m$, the same result gives only finitely many $n$ satisfying $\phi (n)=m$. The union of these finitely many finite sets is finite, proving the claim.