IT-CESENATICO 2019 Problem 2

Let $p,q$ be prime numbers $.$ Prove that if $p+q^2$ is a perfect square $,$ then $p^2+q^n$ is not a perfect square for any positive integer $n.$

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.