BALTIC-WAY 2011 Problem 18

Determine all pairs $(p,q)$ of primes for which both $p^2+q^3$ and $q^2+p^3$ are perfect squares.

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.