SILK-ROAD 2004 Problem 2

find all primes $p$ , for which exist natural numbers, such that $p=m^2+n^2$ and $p|(m^3+n^3-4)$ .

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.