Integer points on the curve 𝑌²=𝑋³±𝑝^{𝑘}𝑋

Draziotis, Konstantinos

doi:10.1090/S0025-5718-06-01852-7

Integer points on the curve $Y^{2}=X^{3}\pm p^{k}X$
HTML articles powered by AMS MathViewer

by Konstantinos A. Draziotis PDF

Math. Comp. 75 (2006), 1493-1505 Request permission

Abstract:

We completely solve diophantine equations of the form $Y^{2}=X^{3}\pm p^{k}X,$ where $k$ is a positive integer, using a reduction to some quartic elliptic equations, which can be solved with well known methods.

References

Michael A. Bennett, On the representation of unity by binary cubic forms, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1507–1534. MR 1806730, DOI 10.1090/S0002-9947-00-02658-1
Jian Hua Chen and Paul Voutier, Complete solution of the Diophantine equation $X^2+1=dY^4$ and a related family of quartic Thue equations, J. Number Theory 62 (1997), no. 1, 71–99. MR 1430002, DOI 10.1006/jnth.1997.2018
J. H. E. Cohn, The Diophantine equation $y^{2}=Dx^{4}+1$. III, Math. Scand. 42 (1978), no. 2, 180–188. MR 512268, DOI 10.7146/math.scand.a-11746
J. H. E. Cohn, The Diophantine equation $x^4+1=Dy^2$, Math. Comp. 66 (1997), no. 219, 1347–1351. MR 1415800, DOI 10.1090/S0025-5718-97-00851-X
Kevin R. Coombes and David R. Grant, On heterogeneous spaces, J. London Math. Soc. (2) 40 (1989), no. 3, 385–397. MR 1053609, DOI 10.1112/jlms/s2-40.3.385
Sinnou David, Minorations de formes linéaires de logarithmes elliptiques, Mém. Soc. Math. France (N.S.) 62 (1995), iv+143 (French, with English and French summaries). MR 1385175
J. Gebel, A. Pethő, and H. G. Zimmer, Computing integral points on elliptic curves, Acta Arith. 68 (1994), no. 2, 171–192. MR 1305199, DOI 10.4064/aa-68-2-171-192
Genocchi, Sur l’impossibilite de quelques egalites doubles, C. R. Acad.Sci. Paris, 78 (1874), 423-436.
Aleksander Grytczuk, Florian Luca, and Marek Wójtowicz, The negative Pell equation and Pythagorean triples, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), no. 6, 91–94. MR 1769976
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
Wilhelm Ljunggren, Zur Theorie der Gleichung $x^2+1=Dy^4$, Avh. Norske Vid.-Akad. Oslo I 1942 (1942), no. 5, 27 (German). MR 16375
—, Einige Eigenschften der Einheiten reel Quadratischer und rein-bi-quadratischer Zahlkorper. Skr. Norske Vid. Akad. Oslo I, v.1936, no.12.
F. Luca and P. G. Walsh, A generalization of a theorem of Cohn on the equation $x^3-Ny^2=\pm 1$, Rocky Mountain J. Math. 31 (2001), no. 2, 503–509. MR 1840950, DOI 10.1216/rmjm/1020171571
L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
L. J. Mordell, The Diophantine equation $y^{2}=Dx^{4}+1$, J. London Math. Soc. 39 (1964), 161–164. MR 162761, DOI 10.1112/jlms/s1-39.1.161
Dimitrios Poulakis, A simple method for solving the Diophantine equation $Y^2=X^4+aX^3+bX^2+cX+d$, Elem. Math. 54 (1999), no. 1, 32–36 (English, with German summary). MR 1669371, DOI 10.1007/s000170050053
D. Poulakis and P. G. Walsh, A note on the Diophantine equation $X^2-dY^4=1$ with prime discriminant, C. R. Math. Acad. Sci. Soc. R. Can. 27 (2005), no. 2, 54–57 (English, with English and French summaries). MR 2142959
H. E. Rose, A course in number theory, 2nd ed., Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1352868
Pierre Samuel, Résultats élémentaires sur certaines équations diophantiennes, J. Théor. Nombres Bordeaux 14 (2002), no. 2, 629–646 (French, with English and French summaries). MR 2040698
A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958), 185–208; erratum 5 (1958), 259 (French). MR 106202, DOI 10.4064/aa-4-3-185-208
W. Sierpiński, Elementary theory of numbers, 2nd ed., North-Holland Mathematical Library, vol. 31, North-Holland Publishing Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw, 1988. Edited and with a preface by Andrzej Schinzel. MR 930670
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
N. P. Smart, $S$-integral points on elliptic curves, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 3, 391–399. MR 1291748, DOI 10.1017/S0305004100072698
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, DOI 10.4064/aa-67-2-177-196
Roel J. Stroeker and Nikos Tzanakis, On the elliptic logarithm method for elliptic Diophantine equations: reflections and an improvement, Experiment. Math. 8 (1999), no. 2, 135–149. MR 1700575
A. Togbe, P. M. Voutier, and P. G. Walsh, Solving a family of Thue equations with an application to the equation $x^2-Dy^4=1$, Acta Arith. 120 (2005), no. 1, 39–58. MR 2189717, DOI 10.4064/aa120-1-3
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, DOI 10.1016/0022-314X(89)90014-0
Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
P. G. Walsh, Diophantine equations of the form $aX^4-bY^2=\pm 1$, Algebraic number theory and Diophantine analysis (Graz, 1998) de Gruyter, Berlin, 2000, pp. 531–554. MR 1770484
Gary Walsh, A note on a theorem of Ljunggren and the Diophantine equations $x^2-kxy^2+y^4=1,4$, Arch. Math. (Basel) 73 (1999), no. 2, 119–125. MR 1703679, DOI 10.1007/s000130050376
Don Zagier, Large integral points on elliptic curves, Math. Comp. 48 (1987), no. 177, 425–436. MR 866125, DOI 10.1090/S0025-5718-1987-0866125-3