KöMaL 2016 B4786

Let $p$ and $q$ be positive integers. Prove that at least one of $p^2+q$ and $p+q^2$ is not a perfect square.

Official text KöMaL. 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.