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Large integral points on elliptic curves

Author: Don Zagier
Journal: Math. Comp. 48 (1987), 425-436
MSC: Primary 11G05; Secondary 11D25, 11Y50
Addendum: Math. Comp. 51 (1988), 375.
MathSciNet review: 866125
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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 $ {y^2} = {x^3} + ax + b\;(a,b \in {\mathbf{Z}})$ in a large range $ \vert x\vert,\vert y\vert \leqslant B$, in time polynomial in $ \log \log B$. We also give a number of individual examples and of parametric families of examples of specific elliptic curves having a relatively large integral point.

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  • [1] A. Baker, Transcendental Number Theory, Cambridge Univ. Press, Cambridge, 1975. MR 0422171 (54:10163)
  • [2] W. Borho, Befreundete Zahlen, Ein zweitausend Jahre altes Thema der elementaren Zahlentheorie, Mathematische Miniaturen, Birkhäuser, Basel, 1981, pp. 5-38. MR 643809 (83k:10017)
  • [3] J. P. Buhler, B. H. Gross & D. B. Zagier, "On the conjecture of Birch and Swinnerlon-Dyer for an elliptic curve of rank 3," Math. Comp., v. 44, 1985, pp. 473-481. MR 777279 (86g:11037)
  • [4] J. H. Conway & N. J. A. Sloane, "Lorentzian forms for the Leech lattice," Bull. Amer. Math. Soc. (NS.), v. 6, 1982, pp. 215-217. MR 640949 (84d:10036)
  • [5] H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917.
  • [6] W. Ellison, F. Ellison, J. Pesek, C. Stahl & D. Stall, "The Diophantine equation $ {y^2} + k = {x^3}$," J. Number Theory, v. 4, 1972, pp. 107-117. MR 0316376 (47:4923)
  • [7] A. K. Lenstra, H. W. Lenstra, Jr. & L. Lovász, "Factoring polynomials with rational coefficients," Math. Ann., v. 261, 1982, pp. 515-534. MR 682664 (84a:12002)
  • [8] J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Text 106, Springer, New York, 1986. MR 817210 (87g:11070)
  • [9] R. P. Steiner, "On Mordell's equation $ {y^2} - k = {x^3}$: A problem of Stolarsky," Math. Comp., v. 46, 1986, pp. 703-714. MR 829640 (87e:11041)

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Article copyright: © Copyright 1987 American Mathematical Society

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