Solutions to a generalized Fermat equation that has even exponents

The equation x2m+y2n = z2r is considered for positive integer values of m, n, and r. If m, n, and r are not pairwise relatively prime, then there are no solutions to this equation in nonzero integers. Formulas that generate infinitely many solutions to this equation are given when m, n, and r are pairwise relatively prime; the newly discovered formulas tend to yield non-primitive solutions. Along the way, it is shown that x2 + y2n = z2n (or equivalently, z2n − y2n = x2) has no solutions in nonzero integers when n > 1. Mathematics Subject Classification: 11D41