Practical solution of the Diophantine equation 𝑦²=𝑥(𝑥+2^{𝑎}𝑝^{𝑏})(𝑥-2^{𝑎}𝑝^{𝑏})

Draziotis, Konstantinos; Poulakis, Dimitrios

doi:10.1090/S0025-5718-06-01841-2

Practical solution of the Diophantine equation

Authors: Konstantinos Draziotis and Dimitrios Poulakis
Journal: Math. Comp. 75 (2006), 1585-1593
MSC (2000): Primary 11Y50; Secondary 11D25, 11G05
DOI: https://doi.org/10.1090/S0025-5718-06-01841-2
Published electronically: March 29, 2006
MathSciNet review: 2219047
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Abstract: Let be an odd prime and , positive integers. In this note we prove that the problem of the determination of the integer solutions to the equation can be easily reduced to the resolution of the unit equation $u+\sqrt{2}v = 1$ over $\mathbb{Q}(\sqrt{2},\sqrt{p})$ . The solutions of the latter equation are given by Wildanger's algorithm.

1. Michael A. Bennett and Gary Walsh, The Diophantine equation 𝑏²𝑋⁴-𝑑𝑌²=1, Proc. Amer. Math. Soc. 127 (1999), no. 12, 3481–3491. MR 1625772, https://doi.org/10.1090/S0002-9939-99-05041-8
2. A. Bremner, J. H. Silverman, and N. Tzanakis, Integral points in arithmetic progression on 𝑦²=𝑥(𝑥²-𝑛²), J. Number Theory 80 (2000), no. 2, 187–208. MR 1740510, https://doi.org/10.1006/jnth.1999.2430
3. Yann Bugeaud, On the size of integer solutions of elliptic equations, Bull. Austral. Math. Soc. 57 (1998), no. 2, 199–206. MR 1617363, https://doi.org/10.1017/S0004972700031592
4. Claude Chabauty, Démonstration de quelques lemmes de rehaussement, C. R. Acad. Sci. Paris 217 (1943), 413–415 (French). MR 11571
5. Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206
6. J. H. E. Cohn, The Diophantine equation 𝑥⁴-𝐷𝑦²=1. II, Acta Arith. 78 (1997), no. 4, 401–403. MR 1438594, https://doi.org/10.4064/aa-78-4-401-403
7. J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
8. K. Draziotis, Integral solutions of the equation $Y^2 = X^3\pm p^kX$ , Math. Comp. (to appear).
9. Keqin Feng and Maosheng Xiong, On elliptic curves 𝑦²=𝑥³-𝑛²𝑥 with rank zero, J. Number Theory 109 (2004), no. 1, 1–26. MR 2098473, https://doi.org/10.1016/j.jnt.2003.12.015
10. Josef Gebel and Horst G. Zimmer, Computing the Mordell-Weil group of an elliptic curve over 𝐐, Elliptic curves and related topics, CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994, pp. 61–83. MR 1260955
11. J. Gebel, A. Pethő, and H. G. Zimmer, Computing integral points on elliptic curves, Acta Arith. 68 (1994), no. 2, 171–192. MR 1305199, https://doi.org/10.4064/aa-68-2-171-192
12. G. Hanrot, Resolution effective d'equations diophantiennes: algorithmes et applications, These, Universite Bordeaux 1 (1997).
13. Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, New York-Berlin, 1980. Reprint of the 1974 original. MR 600654
14. Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. MR 766911
15. Serge Lang, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 231, Springer-Verlag, Berlin-New York, 1978. MR 518817
16. L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
17. Dimitrios Poulakis, Integer points on algebraic curves with exceptional units, J. Austral. Math. Soc. Ser. A 63 (1997), no. 2, 145–164. MR 1475559
18. Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210
19. N. P. Smart, 𝑆-integral points on elliptic curves, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 3, 391–399. MR 1291748, https://doi.org/10.1017/S0305004100072698
20. R. J. Stroeker and N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67 (1994), no. 2, 177–196. MR 1291875, https://doi.org/10.4064/aa-67-2-177-196
21. N. Tzanakis and B. M. M. de Weger, On the practical solution of the Thue equation, J. Number Theory 31 (1989), no. 2, 99–132. MR 987566, https://doi.org/10.1016/0022-314X(89)90014-0
22. K. Wildanger, Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern, J. Number Theory 82 (2000), no. 2, 188–224 (German, with English summary). MR 1761620, https://doi.org/10.1006/jnth.1999.2414
23. http://magma.maths.usyd.edu.au/magma/.