There is a simple criterion for the existence of nontrivial solutions in terms of congruences modulo a , b and c . Similarly with square factors of b or c . From now on we assume that a , b , c are pairwise coprime and square-free. Theorem 4.9 Let be pairwise coprime and square-free. When there is one solution then the complete set of solutions may be parametrized as with Pythagorean Triples, expressing each of x , y , z as a homogeneous quadratic in two parameters u , v .
Solutions exist since the congruences , and all have solutions.
There are much faster ways of solving the equation in practice than just searching, but our proof does show that a search of approximately abc triples will certainly produce a solution.