New exponents for the generalized Fermat equation x² + y³ = zⁿ

We determine the primitive solutions of x² + y³ = zⁿ for n = 11, 17 and 19 unconditionally; for n = 11 Freitas, Naskręcki and Stoll did so under the generalized Riemann hypothesis (GRH). For n = 25 we do not assume GRH but use a reduction of Freitas and Stoll that is available as a preprint, and for n = 13, 23, 26 and 29 we assume GRH. GRH enters only through the class numbers of eight fields of degree 24 and 30 and four of degree 56. The solutions are the trivial ones, with xyz = 0, and those coming from 3² − 2³ = 1. With earlier work this settles every exponent 6 ≤ n ≤ 36 except 31, under the same conditions. Following Freitas, Naskręcki and Stoll, and Freitas and Stoll for n = 25, we reduce to points over number fields on modular curves or on curves Y² = X⁵ + β. The new step is to find these points by descents that need only the class groups of their fields of definition or of small extensions, mostly along isogenies with kernel μ_ℓ, along 1 − σ for a diamond automorphism σ, and along 1 − ζ₅.