Home > Olympiad > IMC 2011 Day 1 Problem 3

## IMC 2011 Day 1 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 $f,g \in \Bbb{Z}[X]$.

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

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

Again, it is not a complete answer. Indeed, for $p=2$ we have that $p^p-1$ is the minimal interesting number. The idea is to find all prime numbers $p$ such that $p^p-1$ is the minimal interesting number when $p$ is fixed.