KMO 2012 Problem 3

Find all triples $(m,p,q)$ where $m$ is a positive integer and $p , q$ are primes. $2^m p^2 + 1 = q^5$

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.