Calculate, for $a, b \in \mathbb{N}^{*}$ coprime, and $m, n \in \mathbb{N}^{*}, \operatorname{gcd}\left(a^{n}-b^{n}, a^{m}-b^{m}\right)$.
Let $d$ be a common divisor of $a^{m}-b^{m}$ and $a^{n}-b^{n}$. Evidently, $d$ is coprime with $a$ and with $b$. Therefore, in $\mathbb{Z} / d \mathbb{Z}$, we can consider negative powers of $a$ and $b$. Let $u, v \in \mathbb{Z}$ such that $n u + m v = m \wedge n$. We have $a^{n} \equiv b^{n} \pmod{d}$, so $a^{n u} \equiv b^{n u} \pmod{d}$; we have $a^{m} \equiv b^{m} \pmod{d}$, so $a^{m v} \equiv b^{m v} \pmod{d}$, and consequently $a^{m \wedge n} = a^{m v + n u} \equiv a^{m v} a^{n u} \equiv b^{m v} b^{n u} \equiv b^{m v + n u} = a^{m \wedge n}$ (modulo $d$), and $d \mid a^{m \wedge n} - b^{m \wedge n}$.

Conversely, $a^{m \wedge n} - b^{m \wedge n}$ divides $a^{m} - b^{m}$ and $a^{n} - b^{n}$. The gcd sought is therefore $\left|a^{m \wedge n} - b^{m \wedge n}\right|$.