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

DOI:
https://doi.org/10.1090/S0002-9939-03-07311-8

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 -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:
https://doi.org/10.1090/S0002-9939-03-07311-8

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