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RECURRENCE RELATIONS FOR ELLIPTIC SEQUENCES: EVERY SOMOS 4 IS A SOMOS LOWERCASE $k$

Published online by Cambridge University Press:  24 July 2006

ALFRED J. VAN DER POORTEN  and
CHRISTINE S. SWART
ALFRED J. VAN DER POORTEN
Affiliation:
Centre for Number Theory Research, 1 Bimbil Place, Killara, Sydney 2071, Australiaalf@math.mq.edu.au
CHRISTINE S. SWART
Affiliation:
Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 South Africacswart@maths.uct.ac.za
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Abstract

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 $\sigma$-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 $5, 6, 7, \ldots.$

Keywords

11B8311G05 (primary)11A5514H0514H52 (secondary)
Type
Papers
Information
Bulletin of the London Mathematical Society , Volume 38 , Issue 4 , August 2006 , pp. 546 - 554
Copyright
The London Mathematical Society 2006

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