On powers as sums of two cubes

Abstract

In a paper of Kraus, it is proved that x3 + y3 = zp for p ≥ 17 has only trivial primitive solutions, provided that p satisfies a relatively mild and easily tested condition. In this article we prove that the primitive solutions of x3 + y3 = zp with p = 4; 5; 7; 11; 13, correspond to rational points on hyperelliptic curves with Jacobians of relatively small rank. Consequently, Chabauty methods may be applied to try to find all rational points. We do this for p = 4; 5, thus proving that x3 + y3 = z4 and x3 + y3 = z5 have only trivial primitive solutions. In the process we meet a Jacobian of a curve that has more 6-torsion at any prime of good reduction than it has globally. Furthermore, some pointers are given to computational aids for applying Chabauty methods.