The generalized Fermat equation X^5+Y^3=Z^7

Assuming the classification of Dahmen–Siksek and Putz, we show that X^5+Y^3=Z^7 has no solution in nonzero coprime integers; equivalently, the generalized Fermat equation of signature (3,5,7) has no nontrivial primitive solution in any arrangement of the exponents. By that classification, a solution determines one of seven septic fields: six pure fields Q((3^a·5)^(1/7)), 1 ≤ a ≤ 6, and an exceptional field F. We exclude F by a fifth-power descent over F(√−35), where the equation becomes a norm identity for an explicit element; the identity and the element's valuations above 3, 5 and 7 confine it to 125 classes modulo fifth powers, none compatible with its possible fifth-power residue symbols above 11 and 19. For the pure fields we use results from recent preprints on the genus-2 Frey curves of Pacetti and Villagra Torcomian, whose Jacobians have real multiplication by Q(√5). Level lowering shows that their 7-torsion is reducible, giving a rational point on a genus-15 double cover of the genus-8 curve of F_49-lines in the 7-torsion. The Prym variety of this cover contains an abelian surface of rank one, and Chabauty's method at 13 confines the point to two explicit rational points, neither coming from a solution. Supplementary data and programs: https://doi.org/10.5281/zenodo.22949370. P. Chocian (arXiv:2609.26996) has proved the same theorem independently, by different methods.