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

Full-text PDF

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.

**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, http://www.maths.nott.ac.uk/ 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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11G05,
11A41

Retrieve articles in all journals with MSC (2000): 11G05, 11A41

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