TWISTS OF X(7) AND PRIMITIVE SOLUTIONS TO x 2 + y 3 = z 7

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. 1.