Glasnik Matematicki, Vol. 52, No. 1 (2017), 1-10.

GEOMETRIC PROGRESSIONS ON ELLIPTIC CURVES

Abdoul Aziz Ciss and Dustin Moody

Laboratoire de Traitement de l'Information et Systèmes Intelligents,, École Polytechnique de Thiès, BP A10 Thiès, Sénégal
e-mail: aaciss@ept.sn

National Institute of Standards and Technology (NIST), 100 Bureau Drive, Gaithersburg, 20899-8930, USA
e-mail: dustin.moody@nist.gov

Abstract.   In this paper, we look at long geometric progressions on different models of elliptic curves, namely Weierstrass curves, Edwards and twisted Edwards curves, Huff curves and general quartics curves. By a geometric progression on an elliptic curve, we mean the existence of rational points on the curve whose x-coordinates (or y-coordinates) are in geometric progression. We find infinite families of twisted Edwards curves and Huff curves with geometric progressions of length 5, an infinite family of Weierstrass curves with 8-term progressions, as well as infinite families of quartic curves containing 10-term geometric progressions.

2010 Mathematics Subject Classification.   11B25, 11D41, 11G05.

Key words and phrases.   Arithmetic progression, geometric progression, elliptic curves.

Full text (PDF) (free access)

DOI: 10.3336/gm.52.1.01

References:

  1. A. Alvarado, An arithmetic progession on quintic curves, J. Integer Seq. 12 (2009), Article 09.7.3, 6pp.
    MathSciNet    

  2. A. Berczes and V. Ziegler, On geometric progressions on Pell equations and Lucas sequences, Glas. Mat. Ser. III 48(68) (2013), 1-22.
    MathSciNet     CrossRef

  3. D. J. Bernstein, P. Birkner, M. Joye, T. Lange and P. Peters, Twisted Edwards curves, in Progress in Cryptology (AFRICACRYPT'08), Springer, Berlin, 2008, 389-405.
    MathSciNet     CrossRef

  4. A. Bremner, On arithmetic progressions on elliptic curves, Experiment. Math. 8 (1999), 409-413.
    MathSciNet     CrossRef     Link

  5. A. Bremner, Arithmetic progressions on Edwards curves, J. Integer Seq. 16 (2013), Article 13.8.5, 5pp.
    MathSciNet    

  6. A. Bremner and M. Ulas, Rational points in geometric progressions on certain hyperelliptic curves, Publ. Math. Debrecen 82 (2013), 669-683.
    MathSciNet     CrossRef

  7. G. Campbell, A note on arithmetic progressions on elliptic curves, J. Integer Seq. 6 (2003), Article 03.1.3, 5pp.
    MathSciNet    

  8. A. Choudhry, Arithmetic progressions on Huff curves, J. Integer Seq. 18 (2015), Article 15.5.2, 9pp.
    MathSciNet    

  9. H. M. Edwards, A normal form for elliptic curves, Bull. Amer. Math. Soc. (N.S.) 44 (2007), 393-422.
    MathSciNet     CrossRef

  10. I. García-Selfa and J. Tornero, Searching for simultaneous arithmetic progressions on elliptic curves, Bull. Austral. Math. Soc. 71 (2005), 417-424.
    MathSciNet     CrossRef

  11. E. González-Jiménez, Markoff-Rosenberger triples in geometric progression, Acta Math. Hungar. 142 (2014), 231-243.
    MathSciNet     CrossRef

  12. E. Gonzalez-Jiménez, On arithmetic progressions on Edwards curves, Acta Arith. 167 (2015), 117-132.
    MathSciNet     CrossRef

  13. G. B. Huff, Diophantine problems in geometry and elliptic ternary forms, Duke Math. J. 15 (1948), 443-453.
    MathSciNet     CrossRef     Link

  14. M. Joye, M. Tibouchi, and D. Vergnaud, Huff's model for elliptic curves, in Algorithmic number theory (ANTS-IX), Springer, Berlin, 2010, 234-250.
    MathSciNet     CrossRef

  15. A. MacLeod, 14-term arithmetic progressions on quartic elliptic curves, J. Integer Seq. 9 (2006), Article 06.1.2, 4pp.
    MathSciNet    

  16. D. Moody, Arithmetic progressions on Edwards curves, J. Integer Seq. 14 (2011), Article 11.1.7, 4pp.
    MathSciNet    

  17. D. Moody, Arithmetic progressions on Huff curves, Ann. Math. Inform. 38 (2011), 111-116.
    MathSciNet    

  18. D. Moody and A. S. Zargar, On the rank of elliptic curves with long arithmetic progressions, to appear in Colloq. Math., 2016.
    CrossRef

  19. Sage software, Version 4.5.3, http://sagemath.org.

  20. J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Springer-Verlag, New-York, 1994.
    MathSciNet     CrossRef

  21. M. Ulas, A note on arithmetic progressions on quartic elliptic curves, J. Integer Seq. 8 (2005), Article 05.3.1, 5 pp.
    MathSciNet    

  22. M. Ulas, On arithmetic progressions on genus two curves, Rocky Mountain J. Math. 39 (2009), 971-980.
    MathSciNet     CrossRef

  23. L. Washington, Elliptic curves: number theory and cryptography, CRC Press, 2008.
    MathSciNet     CrossRef
Glasnik Matematicki Home Page