BG-MO 2016 Problem 1

Find all positive integers $m$ and $n$ such that $(2^{2^{n}}+1)(2^{2^{m}}+1)$ is divisible by $m\cdot n$ .

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.