MathNet 2023-part-04 — 191 problems (EN–VI) Question 8

Let $p \ge 5$ be a prime number and $f(x) = (x-1)(x^3-1)\dots(x^{p-4}-1)(x^{p-2}-1)$. Show that $f(n^k) - f(n)$ is divisible by $p$ for any positive integer $k$ relatively prime to $p-1$.