MathNet 2019-part-07 — 184 problems (EN–VI) Question 56

Let $p$ be an odd prime number, $m \gt 1$ and $n$ be positive integers such that $\frac{m^{pn} - 1}{m^n - 1}$ is a prime number. Show that
$$
pn \mid (p-1)^n + 1.
$$