Mathematics – Number Theory
Scientific paper
2005-08-04
Mathematics
Number Theory
23 pages. Extra examples and results added; further typos corrected
Scientific paper
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. 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.
Hone Andrew
Swart Christine
No associations
LandOfFree
Integrality and the Laurent phenomenon for Somos 4 sequences does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Integrality and the Laurent phenomenon for Somos 4 sequences, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Integrality and the Laurent phenomenon for Somos 4 sequences will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-614797