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On the Integrality of Somos 4 Sequences

Year 2019, , 49 - 55, 24.03.2019
https://doi.org/10.18185/erzifbed.415008

Abstract

In
this paper, it is shown that if (hn) is a Somos 4 sequence
associated to an elliptic curve then each hn can be expressed
as elements of the ring R =
[a1, a2, a3, a4, x, y, h–1±1, h0]. In particular, if h –1 = ±1, then the Somos 4 sequence consist entirely of
integers for n ≥ 0. Also the general term of the Somos 4 sequence is
given which satisfies a binary recurrence relation.

References

  • 1. Everest, G., van der Poorten, A., Shparlinski, I. and Ward, T., (2003). “Recurrence Sequences”, Mathematical Surveys and Monographs 104, AMS, Providence, RI; pp 318.
  • 2. Fomin, S. and Zelevinsky, A. (2002). “The Laurent phenomen”, Adv. Appl. Math., 28, 119-144.
  • 3. Gale, D. (1991). “The strange and suprising saga of the Somos sequences”, Math. Intelligencer, 13 (1) 40-42.
  • 4. Gale, D. (1991). “Somos sequence update”, Math. Intelligencer, 13 (4) 49-50.
  • 5. Gezer, B., Çapa, B. and Bizim, O. (2016), ”A family of integer Somos sequences”, Mathematical Reports, 18 (68), 3 417-435.
  • 6. Hone, A. N. W. (2005), “Elliptic curves and quadratic recurrence sequences”, Bull. Lond. Math. Soc., 37, 161--171; Corrigendum 38 (2006), 741--742.
  • 7. Hone, A. N. W. and Swart, C. (2008). “Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences”, Math. Proc. of the Cambridge Phil. Soc., 145 65-85.
  • 8. Husemöller, D. (1987). “Elliptic Curves”, Springer Verlag, New York,; pp 487.
  • 9. Malouf, J. L. (1992). “An integer sequence from a rational recursion”, Discrete Math., 110 257-261.
  • 10. Propp, J, www.faculty.uml.edu/jpropp/somos.html, (accessed 09. 04.2018).
  • 11. van der Poorten, A. J. (2005). “Elliptic curves and continued fractions”, J. Int. Seq., 8, Article 05.2.5.
  • 12. van der Poorten, A. J. (2006). “Hyperelliptic curves, continued fractions, and Somos sequences”, IMS Lecture Notes-Monograph Series. Dyna-mics & Stochastics, 48 212-224.
  • 13. Robinson, R. (1992). “Periodicity of Somos sequences”, Proc. Amer. Math. Soc., 116 613-619.
  • 14. Silverman, J. H. (2009). “The Arithmetic of Elliptic Curves” 2nd Edition, Graduate Texts in Mathematics 106, Springer Dordrecht Heidelberg London New York; pp 513.
  • 15. Silverman, J. H. And Tate, J. (1992). “Rational Points on Elliptic Curves”, Undergraduate Texts in Mathematics, Springer; pp 281
  • 16. Swart, C. S. (2003). “Elliptic curves and related sequences”, Ph. D. thesis, Royal Holloway (University of London), pp 3-223.
  • 17. Ward, M. (1948). “The law of repetition of primes in an elliptic divisibility sequences”, Duke Math. J., 15 941-946.
  • 18. Ward, M. (1948). “Memoir on elliptic divisibility sequences”, Amer. J. Math., 70 31-74.

Somos 4 Dizilerinin Tamsayılık Özelliği Üzerine

Year 2019, , 49 - 55, 24.03.2019
https://doi.org/10.18185/erzifbed.415008

Abstract



Bu çalışmada,
eğer (hn),
bir eliptik eğri ile eşleşen bir Somos 4 dizisi ise her bir hn
teriminin R =
Z[a1,
a2,
a3,
a4, x, y, h–1±1, h0] halkasının bir
elemanı olarak ifade edilebileceği gösterilmiştir. Özel olarak n ≥ 0
için h –1 =
±1 ise Somos 4 dizisinin terimleri
birer tamsayıdır. Üstelik bir ikili indirgeme bağıntısını gerçekleyen Somos 4
dizisinin genel terimi verilmiştir.


References

  • 1. Everest, G., van der Poorten, A., Shparlinski, I. and Ward, T., (2003). “Recurrence Sequences”, Mathematical Surveys and Monographs 104, AMS, Providence, RI; pp 318.
  • 2. Fomin, S. and Zelevinsky, A. (2002). “The Laurent phenomen”, Adv. Appl. Math., 28, 119-144.
  • 3. Gale, D. (1991). “The strange and suprising saga of the Somos sequences”, Math. Intelligencer, 13 (1) 40-42.
  • 4. Gale, D. (1991). “Somos sequence update”, Math. Intelligencer, 13 (4) 49-50.
  • 5. Gezer, B., Çapa, B. and Bizim, O. (2016), ”A family of integer Somos sequences”, Mathematical Reports, 18 (68), 3 417-435.
  • 6. Hone, A. N. W. (2005), “Elliptic curves and quadratic recurrence sequences”, Bull. Lond. Math. Soc., 37, 161--171; Corrigendum 38 (2006), 741--742.
  • 7. Hone, A. N. W. and Swart, C. (2008). “Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences”, Math. Proc. of the Cambridge Phil. Soc., 145 65-85.
  • 8. Husemöller, D. (1987). “Elliptic Curves”, Springer Verlag, New York,; pp 487.
  • 9. Malouf, J. L. (1992). “An integer sequence from a rational recursion”, Discrete Math., 110 257-261.
  • 10. Propp, J, www.faculty.uml.edu/jpropp/somos.html, (accessed 09. 04.2018).
  • 11. van der Poorten, A. J. (2005). “Elliptic curves and continued fractions”, J. Int. Seq., 8, Article 05.2.5.
  • 12. van der Poorten, A. J. (2006). “Hyperelliptic curves, continued fractions, and Somos sequences”, IMS Lecture Notes-Monograph Series. Dyna-mics & Stochastics, 48 212-224.
  • 13. Robinson, R. (1992). “Periodicity of Somos sequences”, Proc. Amer. Math. Soc., 116 613-619.
  • 14. Silverman, J. H. (2009). “The Arithmetic of Elliptic Curves” 2nd Edition, Graduate Texts in Mathematics 106, Springer Dordrecht Heidelberg London New York; pp 513.
  • 15. Silverman, J. H. And Tate, J. (1992). “Rational Points on Elliptic Curves”, Undergraduate Texts in Mathematics, Springer; pp 281
  • 16. Swart, C. S. (2003). “Elliptic curves and related sequences”, Ph. D. thesis, Royal Holloway (University of London), pp 3-223.
  • 17. Ward, M. (1948). “The law of repetition of primes in an elliptic divisibility sequences”, Duke Math. J., 15 941-946.
  • 18. Ward, M. (1948). “Memoir on elliptic divisibility sequences”, Amer. J. Math., 70 31-74.
There are 18 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Makaleler
Authors

Betül Gezer

Publication Date March 24, 2019
Published in Issue Year 2019

Cite

APA Gezer, B. (2019). On the Integrality of Somos 4 Sequences. Erzincan University Journal of Science and Technology, 12(1), 49-55. https://doi.org/10.18185/erzifbed.415008