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Recurrence Relations for Elliptic Sequences: Every Somos 4 is a Somos k
Author(s) -
Van Der Poorten Alfred J.,
Swart Christine S.
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609306018534
Subject(s) - mathematics , divisibility rule , sequence (biology) , extension (predicate logic) , mathematics subject classification , coherence (philosophical gambling strategy) , quotient , pure mathematics , completeness (order theory) , elliptic function , discrete mathematics , mathematical analysis , statistics , genetics , computer science , biology , programming language
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, …. 2000 Mathematics Subject Classification 11B83, 11G05 (primary), 11A55, 14H05, 14H52 (secondary).