Glasnik Matematicki, Vol. 54, No. 1 (2019), 53-64.

RATIONAL SEQUENCES ON DIFFERENT MODELS OF ELLIPTIC CURVES

Gamze Savaş Çelik, Mohammad Sadek and Gökhan Soydan

Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey
e-mail: gamzesavascelik@gmail.com

Faculty of Engineering and Natural Sciences, Sabancı University, 34956 Tuzla, Istanbul, Turkey
e-mail: mmsadek@sabanciuniv.edu

Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey
e-mail: gsoydan@uludag.edu.tr


Abstract.   Given a set S of elements in a number field k, we discuss the existence of planar algebraic curves over k which possess rational points whose x-coordinates are exactly the elements of S. If the size |S| of S is either 4,5, or 6, we exhibit infinite families of (twisted) Edwards curves and (general) Huff curves for which the elements of S are realized as the x-coordinates of rational points on these curves. This generalizes earlier work on progressions of certain types on some algebraic curves.

2010 Mathematics Subject Classification.   11D25, 11G05, 14G05

Key words and phrases.   Elliptic curve, Edwards curve, Huff curve, rational sequence, rational point


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DOI: 10.3336/gm.54.1.04


References:

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

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

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

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

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

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

  7. G. S. Çelik and G. Soydan, Elliptic curves containing sequences of consecutive cubes, Rocky Mountain J. Math. 48 (2018), 2163-2174.
    MathSciNet     CrossRef

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

  9. M. Kamel and M. Sadek, On sequences of consecutive squares on elliptic curves, Glas. Mat. Ser. III 52(72) (2017), 45-52.
    MathSciNet     CrossRef

  10. J.-B. Lee and W. Y. Vélez, Integral solutions in arithmetic progression for y2=x3+k, Period. Math. Hungar. 25 (1992), 31-49.
    MathSciNet     CrossRef

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

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

  13. A. A. Ciss and D. Moody, Geometric progressions on elliptic curves, Glas. Mat. Ser. III 52(72) (2017), 1-10.
    MathSciNet     CrossRef

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

  15. J. H. Silverman, The arithmetic of elliptic curves, Springer, Dordrecht, 2009.
    MathSciNet     CrossRef

  16. M. Stoll and J. E. Cremona, Minimal models for 2-coverings of elliptic curves, LMS J. Comput. Math. 5 (2002), 220-243.
    MathSciNet     CrossRef

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

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

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