SE-SKOLORNAS 2017 Problem 2

Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $\frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.