The generalized Fermat equation x^2+y^5=z^9

We show that the equation x^2+y^5=z^9 has no solution in coprime nonzero integers. Equivalently, none of the equations x^2+y^5=z^9, x^2+y^9=z^5 and x^5+y^9=z^2 has such a solution. By Edwards' parametrization of the coprime solutions of a^2+b^3+c^5=0, a solution gives coprime integers u, v at which one of 27 binary forms g_i of degree 20 takes the value -z^3. A 2-adic argument and a descent modulo cubes over number fields of degree at most 10 exclude 22 of the 27 forms and, for each of the other five, determine the values of the factors of g_i up to cubes. These five forms lead to rational points on an elliptic curve over Q, on a curve of genus 2 over Q, and on elliptic curves over Q(5^(1/3)) and over Q(sqrt(-5)). We determine the relevant points by a 2-descent over Q, by elliptic Chabauty over a cubic field together with a Mordell-Weil sieve, by a descent via 3-isogeny over Q(5^(1/3)), and by elliptic Chabauty over Q(sqrt(-5)). No step assumes the generalized Riemann hypothesis. Programs and data: https://doi.org/10.5281/zenodo.23004600.