Integrality and the Laurent phenomenon for Somos 4 sequences
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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2005
 arXiv:
 arXiv:math/0508094
 Bibcode:
 2005math......8094S
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics
 EPrint:
 23 pages. Extra examples and results added