TUYMAADA 2000 Problem 5

Are there prime $p$ and $q$ larger than $3$ , such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$ ?

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.