Problem

Source: Ukraine TST 2014 p9

Tags: number theory, prime numbers, parameterization, Diophantine Equations, diophantine, system of equations



Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.