Large integral points on elliptic curves
Author:
Don Zagier
Journal:
Math. Comp. 48 (1987), 425-436
MSC:
Primary 11G05; Secondary 11D25, 11Y50
DOI:
https://doi.org/10.1090/S0025-5718-1987-0866125-3
Addendum:
Math. Comp. 51 (1988), 375.
MathSciNet review:
866125
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We describe several methods which permit one to search for big integral points on certain elliptic curves, i.e., for integral solutions (x, y) of certain Diophantine equations of the form in a large range
, in time polynomial in
. We also give a number of individual examples and of parametric families of examples of specific elliptic curves having a relatively large integral point.
- [1] Alan Baker, Transcendental number theory, Cambridge University Press, London-New York, 1975. MR 0422171
- [2] Walter Borho, Befreundete Zahlen: ein zweitausend Jahre altes Thema der elementaren Zahlentheorie, Living numbers, Math. Miniaturen, vol. 1, Birkhäuser, Basel-Boston, Mass., 1981, pp. 5–38 (German). MR 643809
- [3] Joe P. Buhler, Benedict H. Gross, and Don B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3, Math. Comp. 44 (1985), no. 170, 473–481. MR 777279, https://doi.org/10.1090/S0025-5718-1985-0777279-X
- [4] J. H. Conway and N. J. A. Sloane, Lorentzian forms for the Leech lattice, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 2, 215–217. MR 640949, https://doi.org/10.1090/S0273-0979-1982-14985-0
- [5] H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917.
- [6] W. J. Ellison, F. Ellison, J. Pesek, C. E. Stahl, and D. S. Stall, The Diophantine equation 𝑦²+𝑘=𝑥³, J. Number Theory 4 (1972), 107–117. MR 316376, https://doi.org/10.1016/0022-314X(72)90058-3
- [7] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664, https://doi.org/10.1007/BF01457454
- [8] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210
- [9] Ray P. Steiner, On Mordell’s equation 𝑦²-𝑘=𝑥³: a problem of Stolarsky, Math. Comp. 46 (1986), no. 174, 703–714. MR 829640, https://doi.org/10.1090/S0025-5718-1986-0829640-3
Retrieve articles in Mathematics of Computation with MSC: 11G05, 11D25, 11Y50
Retrieve articles in all journals with MSC: 11G05, 11D25, 11Y50
Additional Information
DOI:
https://doi.org/10.1090/S0025-5718-1987-0866125-3
Article copyright:
© Copyright 1987
American Mathematical Society