Recurrence Relations for Elliptic Sequences : Every Somos 4 Is a Somos K

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 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 gap 4 always also is given by a three term Somos relation of all larger gaps 5, 6 , 7 , . . . . 1. Morgan Ward’s elliptic sequences. In his ‘Memoir on elliptic divisibility sequences’ [10], MorganWard in effect (thus, for all practical purposes) defines antisymmetric double-sided sequences (Wh), that is with W−h = −Wh , by requiring that, for all integers h , m , and n , (1) Wh−mWh+mW 2 n +Wn−hWn+hW 2 m +Wm−nWm+nW 2 h = 0 . If one dislikes double-sided sequences then one rewrites (1) less elegantly as (1) Wh−mWh+mW 2 n = Wh−nWh+nW 2 m −Wm−nWm+nW 2 h , just for h ≥ m ≥ n . In any case, (1) seems more dramatic than it is. An easy exercise confirms that if W1 = 1 then (1) is equivalent to just (2) Wh−mWh+m = W 2 mWh−1Wh+1 −Wm−1Wm+1W 2 h for all integers h ≥ m . Indeed, (2) is just a special case of (1). However, given (2), obvious substitutions in (1) quickly show one may return from (2) to the apparently more general (1). But there is a drama here. The recurrence relation Wh−2Wh+2 = W 2 2 Wh−1Wh+1 −W1W3W 2 h , and non-zero initial values W1 = 1, W2 , W3 , W4 , already suffices to produce the complete sequence! Thus (2) for all m is entailed by its special case m = 2. We could show directly that the case m = 3 follows, see a remark in [9], or a footnote in the corresponding discussion in [5], but the case m = 4, if done asymetrically as in subsequent remarks of [5], plainly was not worth the effort. Plan B, to look it up, fared little better. In her thesis [6], Rachel Shipsey shyly refers the reader back to Morgan Ward’s memoir [10]; but at first glance Ward Typeset February 1, 2008 [20:45] . 2000 Mathematics Subject Classification. Primary: 11B83, 11G05; Secondary: 11A55, 14H05, 14H52.