Glasnik Matematicki, Vol. 44, No.1 (2009), 83-87.

MAXIMAL RANKS AND INTEGER POINTS ON A FAMILY OF ELLIPTIC CURVES

P. G. Walsh

Department of Mathematics, University of Ottawa, 585 King Edward St., Ottawa, Ontario, K1N-6N5, Canada
e-mail: gwalsh@uottawa.ca


Abstract.   We extend a result of Spearman which provides a sufficient condition for elliptic curves of the form y2 = x3 - px, with p a prime, to have Mordell-Weil rank 2. As in Spearman's work, the condition given here involves the existence of integer points on these curves.

2000 Mathematics Subject Classification.   11G05.

Key words and phrases.   Elliptic curve, prime number.


Full text (PDF) (free access)

DOI: 10.3336/gm.44.1.04


References:

  1. J. S. Chahal, Topics in Number Theory, Plenum Press, New York, 1988.
    MathSciNet

  2. J. H. Chen and P. M. Voutier, A complete solution of the Diophantine equation x2 + 1 = dy4 and a related family of quartic Thue equations, J. Number Theory 62 (1997), 71-99.
    MathSciNet     CrossRef

  3. P. Ingram, Multiples of integer points on elliptic curves, preprint.

  4. T. Kudo and K. Motose, On group structures of some special elliptic curves, Math. J. Okayama Univ. 47 (2005), 81-84.
    MathSciNet

  5. J. H. Silverman, Integer points and the rank of Thue elliptic curves, Invent. Math. 66 (1982), 395-404.
    MathSciNet     CrossRef

  6. J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, New York, 1986.
    MathSciNet

  7. B. K. Spearman, Elliptic curves y2 = x3 - px of rank two, Math. J. Okayama Univ. 49 (2007), 183-184.
    MathSciNet

  8. B. K. Spearman, On the group structure of elliptic curves y2 = x3 - 2px, Int. J. Algebra 1 (2007), 247-250.
    MathSciNet

  9. P. G. Walsh, Integer solutions to the equation y2 = x(x2 ± pk), Rocky Mountain J. Math 38 (2008), 1285-1302.
    MathSciNet     CrossRef


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