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

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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 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]**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 ,"*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 : A problem of Stolarsky,"*Math. Comp.*, v. 46, 1986, pp. 703-714. MR**829640 (87e:11041)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0866125-3

Article copyright:
© Copyright 1987
American Mathematical Society