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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Primes generated by elliptic curves
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by Graham Everest, Victor Miller and Nelson Stephens PDF
Proc. Amer. Math. Soc. 132 (2004), 955-963 Request permission

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.
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Additional Information
  • Graham Everest
  • Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom
  • Email: g.everest@uea.ac.uk
  • Victor Miller
  • Affiliation: Center for Communications Research, Princeton, New Jersey 08540
  • Email: victor@idaccr.org
  • Nelson Stephens
  • Affiliation: Department of Mathematical and Computer Sciences, Goldsmiths College, London SE14 6NW, United Kingdom
  • Email: nelson@gold.ac.uk
  • 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
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 955-963
  • MSC (2000): Primary 11G05, 11A41
  • DOI: https://doi.org/10.1090/S0002-9939-03-07311-8
  • MathSciNet review: 2045409