Twists of X(7) and primitive solutions to x(2)+y(3) = z(7)

摘要

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.