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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References
Batut, C., Belabas, K., Bernardi, D., Cohen, H., Olivier, M.: PARI-GP. Avaliable, from ftp://megrez.math.u-bordeaux.fr/pub/pari
Beukers, F.: The Diophantine equation Ax p + By q = Cz r. Duke Math. J. 91(1), 61–88 (1998)
Bruin, N.: Chabauty Methods and Covering Techniques applied to Generalised Fermat Equations. PhD thesis, Universiteit Leiden (1999)
Bruin, N.: The diophantine equations x 2 ± y 4 = ±z 6 and x 2 + y 8 = z 3. Compositio Math. 118, 305–321 (1999)
Cassels, J.W.S., Flynn, E.V.: Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2. LMS–LNS 230. Cambridge University Press, Cambridge (1996)
Chabauty Sur, C.: les points rationnels des variétés algébriques dont l’irrégularité est supvrieure à la dimension. C. R. Acad. Sci. Paris 212, 1022–1024 (1941)
Daberkow, M., Fieker, C., Klüners, J., Pohst, M., Roegner, K., Schörnig, M., Wildanger, K.: KANT V4. J. Symbolic Comput. 24(3-4), 267–283 (1997), Available from ftp://ftp.math.tu-berlin.de/pub/algebra/Kant/Kash
Darmon, H., Granville, A.: On the equations zm = F( x, y) and Axp + Byq = Czrr. Bull. London Math. Soc. 27(6), 513–543 (1995)
Darmon, H., Merel, L.: Winding quotients and some variants of Fermat’s last theorem. J. Reine Angew. Math. 490, 81–100 (1997)
Flynn, E.V.: A flexible method for applying chabauty’s theorem. Compositio Math-ematica 105, 79–94 (1997)
Kraus, A.: private communication
Kraus, A.: Sur l’équation a 3 + b 3 = c p cp. Experiment. Math. 7(1), 1–13 (1998)
Daniel Mauldin, R.: A generalization of Fermat’s last theorem: the Beal conjecture and prize problem. Notices Amer. Math. Soc. 44(11), 1436–1437 (1997)
Odlyzko, A.M.: Tables of discriminant bounds (1976), available at, http://www.research.att.com/~amo/unpublished/index.html
Odlyzko, A.M.: Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. Sém. Théor. Nombres Bordeaux (2) 2(1), 119–141 (1990)
Poonen, B., Schaefer, E.F.: Explicit descent for jacobians of cyclic covers of the projective line. J. reine angew. Math. 488, 141–188 (1997)
Schaefer, E.F.: Computing a Selmer group of a Jacobian using functions on the curve. Math. Ann. 310(3), 447–471 (1998)
Silverman, J.H.: The Arithmetic of Elliptic Curves. GTM 106. Springer, Heidelberg (1986)
Stoll, M.: On the arithmetic of the curves y 2 = x l + A and their Jacobians. J. Reine Angew. Math. 501, 171–189 (1998)
Stoll, M.: On the arithmetic of the curves y 2 = x l + A, II (1998), available from http://www.math.uiuc.edu/Algebraic-Number-Theory
Stoll, M.: Implementing 2-descent for jacobians of hyperelliptic curves (1999), available from http://www.math.uiuc.edu/Algebraic-Number-Theory
Stoll, M.: On the height constant for curves of genus two. Acta Arith. 90(2), 183–201 (1999)
Tijdeman, R.: Diophantine equations and Diophantine approximations. In: Number theory and applications (Banff, AB, 1988), pp. 215–243. Kluwer Acad. Publ., Dordrecht (1989)
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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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