On Powers as Sums of Two Cubes
In a paper of Kraus, it is proved that x 3 + y 3 = z p 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 x 3 + y 3 = z 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.
KeywordsElliptic Curve Rational Point Elliptic Curf Abelian Variety Diophantine Equation
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