The generalized Fermat equation x² + y⁵ = z⁷

We show that the equation x² + y⁵ = z⁷ has no solution in nonzero coprime integers, assuming a classification theorem from Putz's thesis, which has not appeared in a refereed journal. Equivalently, none of the equations x² + y⁵ = z⁷, x² + y⁷ = z⁵, x⁵ + y⁷ = z² has a solution in nonzero coprime integers. Putz attaches to a solution an octic algebra, the fibre of a Belyi map of degree 8, and proves that it is always isomorphic to one fixed octic field L₈. We prove unconditionally that no solution has fibre isomorphic to L₈, by a fifth-power descent over a field L₂₄ of degree 24 that Putz introduced, completed by fifth-power residue symbols at five auxiliary primes. We also prove, without assuming the generalized Riemann hypothesis, that L₂₄ has class number 1, using a criterion of Belabas, Diaz y Diaz and Friedman.