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Sigma function solution of the initial value problem for Somos 5 sequences

“…Zagier's discussion makes use of the inspired substitution (2.16) for the Somos (5) sequence (1.3), although the presentation of the associated curve in the form (2.34) contains a typographical error in [36]. Other approaches to the sequence (1.3) can be found on postings to Propp's "robbins" forum [22]; see, in particular, the references to unpublished work of Elkies in [5], where the sequence (1.3) is related to the problem of Heron triangles with two rational medians. …”

Laurent Polynomials and Superintegrable Maps

“…Based on the appearance of higher-order Somos recurrences in the work of Cantor on the analogues of division polynomials for hyperelliptic curves [6] (see also [40] for analytic formulae), it was conjectured in [27] that every Somos-k sequence should correspond to a discrete linear flow on the Jacobian of such a curve (with an associated discrete integrable system), and the plausibility of this conjecture was justified by a naïve counting argument. However, on Propp's bilinear forum [44], Elkies had already given a more detailed argument to the contrary, based on a proposed theta function formula for the terms of such sequences, which indicated that while the general Somos-6 and Somos-7 sequence could be described by such a formula in genus two, the general Somos-k for k ≥ 8 could not. Thus in this setting the absence of the Laurent property appears to coincide with the absence of algebraic integrability.…”

p-adic properties of division polynomials and elliptic divisibility sequences

“…More recently, elliptic divisibility sequences have been studied by Shipsey [19], who gives an application to the elliptic curve discrete logarithm problem, and by several other authors [3,4,5,8,9,10,23,22]. (See also [11,13,14] for work on the related Somos sequences.) In [22], Nelson Stephens and the author proved that for any fixed modulus 2 e , the sequence…”

“…(See also [11,13,14] Remark 6. Except in some degenerate cases, the definition of an EDS forces W 0 = 0 (put m = n in (38)) and W 1 = ±1 (put n = 1).…”