IZhO 2005 Problem 6

Find all prime numbers $p,q < 2005$ such that $q | p^{2} + 8$ and $p|q^{2} + 8.$

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.