Find all triples $(p,M, z)$ of integers, where $p$ is prime, $m$ is positive and $z$ is negative, that satisfy the equation $$p^3 + pm + 2zm = m^2 + pz + z^2$$
Problem
Source: LCOMA Costa Rica National Olympiad, Final Round, 2015 3.4 day2
Tags: number theory, Diophantine equation, diophantine