Primes generated by elliptic curves
HTML articles powered by AMS MathViewer
- 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
- J. W. S. Cassels, Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, Cambridge, 1991. MR 1144763, DOI 10.1017/CBO9781139172530
- J. E. Cremona, Elliptic Curve Data, up-dated 14-1-02, http://www.maths.nott.ac.uk/ personal/jec/ftp/data/INDEX.html
- 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), no. 4, 385–434. MR 866702, DOI 10.1016/0196-8858(86)90023-0
- Sinnou David, Minorations de formes linéaires de logarithmes elliptiques, Mém. Soc. Math. France (N.S.) 62 (1995), iv+143 (French, with English and French summaries). MR 1385175
- Manfred Einsiedler, Graham Everest, and Thomas Ward, Primes in elliptic divisibility sequences, LMS J. Comput. Math. 4 (2001), 1–13. MR 1815962, DOI 10.1112/S1461157000000772
- 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.
- Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
- Peter Rogers, Topics in Elliptic Divisibility Sequences, MPhil thesis, University of East Anglia, 2003.
- Rachel Shipsey, Elliptic divisibility sequences, Ph.D. thesis, Univ. of London, 2000.
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
- Jacques Vélu, Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A238–A241 (French). MR 294345
- José Felipe Voloch, Siegel’s theorem for complex function fields, Proc. Amer. Math. Soc. 121 (1994), no. 4, 1307–1308. MR 1209430, DOI 10.1090/S0002-9939-1994-1209430-9
- Hermann Kober, Transformationen von algebraischem Typ, Ann. of Math. (2) 40 (1939), 549–559 (German). MR 96, DOI 10.2307/1968939
- Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
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