Recurrence Relations for Elliptic Sequences: every Somos 4 is a Somos $k$

Details Recurrence Relations for Elliptic Sequences: every Somos 4 is a Somos $k$ Recurrence Relations for Elliptic Sequences: every Somos 4 is a Somos $k$

2004-12-15

arxiv.org/abs/math/0412293v1

Mathematics

Number Theory

7 pages

Scientific paper

In his `Memoir on Elliptic Divisibility Sequences', Morgan Ward's definition of the said sequences has the remarkable feature that it does not become at all clear until deep into the paper that there exist nontrivial such sequences. Even then, Ward's proof of coherence of his definition relies on displaying a sequence of values of quotients of Weierstra\ss $\sigma$-functions. We give a direct proof of coherence and show, rather more generally, that a sequence defined by a so-called Somos relation of gap 4 always also is given by a three term Somos relation of all larger gaps 5, 6, 7, $... $.

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