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Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences. (English) Zbl 1165.11018

Somos 4 sequences are defined by a fourth-order quadratic recurrence relation of the form \(A_{n+4} A_n= \alpha A_{n+3} A_{n+1}+ \beta(A_{n+2})^2\), \(\alpha\), \(\beta\) constants. M. Somos, in exploring elliptic theta functions, observed that for \(\alpha= \beta= 1\) and initial values \(A_1= A_2= A_3= A_4= 1\), the sequence consists entirely of integers – surprising in that to find \(A_{n+4}\) one has to divide by \(A_n: 1,1,1, 1,2, 3,7, 23,59,314,\dots\) . Further, similar to the Fibonacci sequence, this one extends backwards:…\(314,59, 23,7, 3,2,1,1,1,1,2,\dots\) . The authors here examine Somos 4 sequences with integer coefficients and initial data and find sufficient conditions for these to be sequences of integers. S. Fomin and A. Zelevinsky [Adv. Appl. Math. 28, No. 2, 119–144 (2002; Zbl 1012.05012)] showed that these recurrences provide one of the simplest examples of the Laurent phenomenon. All the terms of Somos 4 sequences are Laurent polynomials in the initial data.
The authors have established the precise correspondence between Somos 4 sequences and sequences of points on elliptic curves. Further they remark on relationships to Hirota bilinear forms of integrable partial differential equations in soliton theory. Non-periodic sequences provide infinitely many solutions of an associated Diophantine equation in four variables. Analogous results for Somos 5 sequences are also given.

MSC:

11B37 Recurrences
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
94A55 Shift register sequences and sequences over finite alphabets in information and communication theory

Citations:

Zbl 1012.05012
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