On powers as sums of two cubes

In a paper of Kraus, it is proved that x(3) + y(3) = z(p) for p greater than or equal to 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 x(3) + y(3) = x(p) 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 x(3) $ y(3) = z(4) and x(3) + y(3) = z(5) 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.