Find all integers $(a, b)$ such that $a^{5}=5^{b}$
We have the equation $a^{5}=5^{b}$. If $b>0$, then $a$ divides $5^{b}$, so $a$ is a power of 5, and there exists an integer $c$ such that $a=5^{c}$.

The equation can be rewritten as $5^{5 c}=5^{b} \Longleftrightarrow 5 c=b$

The set of solutions is therefore $\left\{\left(5^{c}, 5 c\right), c \in \mathbb{N}\right\}$