TY - JOUR

T1 - Recurrence relations for elliptic sequences

T2 - Every somos 4 is a somos k

AU - Van Der Poorten, Alfred J.

AU - Swart, Christine S.

PY - 2006/8

Y1 - 2006/8

N2 - In his celebrated memoir, Morgan Ward's definition of elliptic divisibility sequences has the remarkable feature that it does not become at all clear until deep into the paper that there exist nontrivial examples of such sequences. Even then, Ward's proof of the coherence of his definition relies on displaying a sequence of values of quotients of Weierstraß σ-functions. We give a direct proof of coherence and show, rather more generally, that a sequence defined by a so-called Somos relation of width 4 is always also given by three-term Somos relations of all larger widths 5, 6, 7, . . . .

AB - In his celebrated memoir, Morgan Ward's definition of elliptic divisibility sequences has the remarkable feature that it does not become at all clear until deep into the paper that there exist nontrivial examples of such sequences. Even then, Ward's proof of the coherence of his definition relies on displaying a sequence of values of quotients of Weierstraß σ-functions. We give a direct proof of coherence and show, rather more generally, that a sequence defined by a so-called Somos relation of width 4 is always also given by three-term Somos relations of all larger widths 5, 6, 7, . . . .

UR - http://www.scopus.com/inward/record.url?scp=33746365091&partnerID=8YFLogxK

U2 - 10.1112/S0024609306018534

DO - 10.1112/S0024609306018534

M3 - Article

VL - 38

SP - 546

EP - 554

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

SN - 0024-6093

IS - 4

ER -