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Uniform estimates for primitive divisors in elliptic divisibility sequences

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Number Theory, Analysis and Geometry

Abstract

Let P be a nontorsion rational point on an elliptic curve E, given by a minimal Weierstrass equation, and write the first coordinate of nP as A n ∕ D n 2, a fraction in lowest terms. The sequence of values D n is the elliptic divisibility sequence (EDS) associated to P. A prime p is a primitive divisor of D n if p divides D n , and p does not divide any earlier term in the sequence. The Zsigmondy set for P is the set of n such that D n has no primitive divisors. It is known that Z is finite. In the first part of the paper we prove various uniform bounds for the size of the Zsigmondy set, including (1) if the j-invariant of E is integral, then the size of the Zsigmondy set is bounded independently of E and P, and (2) if the abc Conjecture is true, then the size of the Zsigmondy set is bounded independently of E and P for all curves and points. In the second part of the paper, we derive upper bounds for the maximum element in the Zsigmondy set for points on twists of a fixed elliptic curve.

Mathematics Subject Classification (2010): 11G05; Secondary 11B37, 14G25, 14H52

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Authors and Affiliations

  1. Department of Pure Mathematics, Colorado State University, Fort Collins, CO, USA, 80523

    Patrick Ingram

  2. Mathematics Department, Brown University, 1917, Providence, RI, 02912, USA

    Joseph H. Silverman

Authors
  1. Patrick Ingram
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  2. Joseph H. Silverman
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Correspondence to Joseph H. Silverman .

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Editors and Affiliations

  1. , Department of Mathematics, Columbia University, Broadway 2990, New York, 10027, New York, USA

    Dorian Goldfeld

  2. , Department of Mathematics, City College of New York, New York, 10031, New York, USA

    Jay Jorgenson

  3. , Department of Mathematics, Yale University, Hillhouse Ave. 10, New Haven, 06520, Connecticut, USA

    Peter Jones

  4. , Department of Mathematics, California Institute of Technology, E. California Blvd. 1200, Pasadena, 91125, California, USA

    Dinakar Ramakrishnan

  5. , Department of Mathematics, University of California at Berkeley, Evans Hall 925, Berkeley, 94720-3840, California, USA

    Kenneth Ribet

  6. , Department of Mathematics, Harvard University, Science Center/One Oxford Street, Cambridge, 02138, Massachusetts, USA

    John Tate

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Dedicated to the memory of Serge Lang

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Ingram, P., Silverman, J.H. (2012). Uniform estimates for primitive divisors in elliptic divisibility sequences. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_12

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