Abstract
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.
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Bruin, N. (2000). On Powers as Sums of Two Cubes. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_9
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DOI: https://doi.org/10.1007/10722028_9
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