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Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences

Published online by Cambridge University Press:  01 July 2008

ANDREW N. W. HONE  and
CHRISTINE SWART
ANDREW N. W. HONE
Affiliation:
Institute of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury CT2 7NF.
CHRISTINE SWART
Affiliation:
Department of Mathematics & Applied Mathematics, University of Cape Town, Rondebosch, 7700, South Africa.
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Abstract

Somos 4 sequences are a family of sequences defined by a fourth-order quadratic recurrence relation with constant coefficients. For particular choices of the coefficients and the four initial data, such recurrences can yield sequences of integers. Fomin and Zelevinsky have used the theory of cluster algebras to prove that these recurrences also provide one of the simplest examples of the Laurent phenomenon: all the terms of a Somos 4 sequence are Laurent polynomials in the initial data. The integrality of certain Somos 4 sequences has previously been understood in terms of the Laurent phenomenon. However, each of the authors of this paper has independently established the precise correspondence between Somos 4 sequences and sequences of points on elliptic curves. This connection is Here we show that these sequences satisfy a stronger condition than the Laurent property, and hence establish a broad set of sufficient conditions for integrality. As a by-product, non-periodic sequences provide infinitely many solutions of an associated quartic Diophantine equation in four variables. The analogous results for Somos 5 sequences are also presented, as well as various examples, including parameter families of Somos 4 integer sequences.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 1 , July 2008 , pp. 65 - 85
Copyright
Copyright © Cambridge Philosophical Society 2008

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