On the Integrality of Somos 4 Sequences

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

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