FI-MO 2013 Problem 5

Find all integer triples $(m,p,q)$ satisfying $2^mp^2+1=q^5$ where $m>0$ and both $p$ and $q$ are prime numbers.

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.