MathNet 2023-part-05 — 191 problems (EN–VI) Question 179

Denote by $\mathbb{N}$ the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $x - y$ divides $x^{f(x)} - y^{f(y)}$ for every two coprime integers $x$ and $y$.