Hone, A.N.W. and Swart, C. (2008) Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences. Mathematical Proceedings of the Cambridge Philosophical Society, 145 (Part I). pp. 6585. ISSN 03050041. (Full text available)
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Official URL http://dx.doi.org/10.1017/s030500410800114x 
Abstract
Somos 4 sequences are a family of sequences defined by a fourthorder 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. 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 byproduct, nonperiodic 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.
Item Type:  Article 

Uncontrolled keywords:  Number Theory (math.NT); Combinatorics (math.CO 
Subjects:  Q Science > QA Mathematics (inc Computing science) 
Divisions:  Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science 
Depositing User:  Jane Griffiths 
Date Deposited:  20 Apr 2009 14:42 
Last Modified:  16 Dec 2011 11:23 
Resource URI:  http://kar.kent.ac.uk/id/eprint/15690 (The current URI for this page, for reference purposes) 
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