Twists of X(7) and primitive solutions to x2+y3=z7

Abstract

We find the primitive integer solutions to $x^2+y^3=z^7$. A nonabelian descent argument involving the simple group of order $168$ reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve $X$. To restrict the set of relevant twists, we exploit the isomorphism between $X$ and the modular curve $X(7)$ and use modularity of elliptic curves and level lowering. This leaves $10$ genus $3$ curves, whose rational points are found by a combination of methods