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Primes generated by elliptic curves

Authors: Graham Everest, Victor Miller and Nelson Stephens
Journal: Proc. Amer. Math. Soc. 132 (2004), 955-963
MSC (2000): Primary 11G05, 11A41
Published electronically: November 7, 2003
MathSciNet review: 2045409
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Abstract | References | Similar Articles | Additional Information

Abstract: For a rational elliptic curve in Weierstrass form, Chudnovsky and Chudnovsky considered the likelihood that the denominators of the $x$-coordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a rational point under an isogeny, we use Siegel's Theorem to prove that only finitely many primes will arise. The same question is considered for elliptic curves in homogeneous form, prompting a visit to Ramanujan's famous taxi-cab equation. Finiteness is provable for these curves with no extra assumptions. Finally, consideration is given to the possibilities for prime generation in higher rank.

References [Enhancements On Off] (What's this?)

  • 1. J. W. S. Cassels,
    Lectures on Elliptic Curves,
    London Mathematical Society Student Texts 24, Cambridge University Press, Cambridge, 1991. MR 92k:11058
  • 2. J. E. Cremona, Elliptic Curve Data, up-dated 14-1-02, personal/jec/ftp/data/INDEX.html
  • 3. D. V. Chudnovsky and G. V. Chudnovsky,
    Sequences of numbers generated by addition in formal groups and new primality and factorization tests,
    Adv. in Appl. Math. 7 (1986), 385-434. MR 88h:11094
  • 4. David Sinnou,
    Minorations de formes linéaires de logarithmes elliptiques,
    Mém. Soc. Math. France (N.S.) (1995), no. 62, iv+143. MR 98f:11078
  • 5. Manfred Einsiedler, Graham Everest and Thomas Ward,
    Primes in elliptic divisibility sequences,
    LMS J. Comput. Math. 4 (2001), 1-13. MR 2002e:11181
  • 6. Graham Everest, Peter Rogers and Thomas Ward,
    A higher rank Mersenne problem,
    ANTS V Proceedings, Springer Lecture Notes in Computer Science, 2369 (2002), 95-107.
  • 7. Marc Hindry and Joseph H. Silverman,
    Diophantine Geometry,
    Graduate Texts in Mathematics, Volume 201, Springer-Verlag, New York, 2000. MR 2001e:11058
  • 8. Peter Rogers,
    Topics in Elliptic Divisibility Sequences,
    MPhil thesis, University of East Anglia, 2003.
  • 9. Rachel Shipsey,
    Elliptic divisibility sequences,
    Ph.D. thesis, Univ. of London, 2000.
  • 10. Joseph H. Silverman,
    The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, No. 106,
    Springer-Verlag, New York, 1986. MR 87g:11070
  • 11. J. Vélu,
    Isogénies entre courbes elliptiques,
    C. R. Acad. Sci. Paris 273 (1971), 238-241. MR 45:3414
  • 12. J. F. Voloch,
    Siegel's theorem for complex function fields,
    Proc. Amer. Math. Soc. 121 (1994), 1307-1308. MR 94j:11052
  • 13. J. F. Voloch,
    Diophantine approximation on abelian varieties in characteristic p,
    Amer. J. of Math. 117 (1995), 1089-1095. MR 96i:11061
  • 14. Morgan Ward,
    Memoir on elliptic divisibility sequences,
    Amer. J. Math. 70 (1948), 31-74. MR 9:332j

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Additional Information

Graham Everest
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom

Victor Miller
Affiliation: Center for Communications Research, Princeton, New Jersey 08540

Nelson Stephens
Affiliation: Department of Mathematical and Computer Sciences, Goldsmiths College, London SE14 6NW, United Kingdom

Received by editor(s): November 22, 2002
Published electronically: November 7, 2003
Additional Notes: Thanks go to John Cremona, Joe Silverman and Felipe Voloch for helpful comments
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2003 American Mathematical Society