KMO 2020 Problem 4

Find a pair of coprime positive integers $(m,n)$ other than $(41,12)$ such that $m^2-5n^2$ and $m^2+5n^2$ are both perfect squares.

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.