Integrality and the Laurent phenomenon for Somos 4 sequences

TL;DR

This paper explores the integrality of Somos 4 sequences, demonstrating they satisfy a stronger property than the Laurent phenomenon, and connects these sequences to elliptic curves and Diophantine equations.

Contribution

The authors establish a stronger condition than the Laurent property for Somos 4 sequences and link them to elliptic curves, providing new integrality criteria and solutions to related Diophantine equations.

Findings

Somos 4 sequences satisfy a stronger property than the Laurent phenomenon.

Sequences correspond to points on elliptic curves, ensuring integrality.

Non-periodic sequences yield infinitely many solutions to a quartic Diophantine equation.

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. 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.