## 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.## References

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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