IMC 2011 · Problem 3

Let $p$ be a prime number. Call a positive integer $n$ interesting if

$$x^n - 1 = (x^p - x + 1)f(x) + pg(x)$$

for some polynomials $f$ and $g$ with integer coefficients.

a) Prove that the number $p^p - 1$ is interesting.

b) For which $p$ is $p^p - 1$ the minimal interesting number?