$n$ is the order of $p \mod p^n-1$, and the order divides every $k$ with $p^k \equiv 1 \mod p^n-1$, so especially $k=\phi(p^n-1)$.

it is in fact trivial ... it's appeared in my course as a consequence of non trivial things. Sorry not to have tought on it.

We used it as an exercise for olympiad french training the year you follow the training Alekk...(or maybe the year after?) Pierre.