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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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: 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

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07311-8
PII: S 0002-9939(03)07311-8
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