On the Integrality of Somos 4 Sequences

Year 2019, Volume: 12 Issue: 1, 49 - 55, 24.03.2019

Betül Gezer

https://doi.org/10.18185/erzifbed.415008

Abstract

In
this paper, it is shown that if (h_n) is a Somos 4 sequence
associated to an elliptic curve then each h_n can be expressed
as elements of the ring R = ℤ[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.

Somos 4 Dizilerinin Tamsayılık Özelliği Üzerine

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.

Keywords

Somos 4 sequences, elliptic curves

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.