On Somos-4 and Somos-5 sequences (Q2666617)

The Somos sequences are a class of non-linear integer recurrence sequences associated to each \(k\in\mathbb{N}\) introduced by \textit{M. Somos} [Crux Mathematicorum 15, p. 208 (1989)]. They have multiple connections to combinatorics and the theory of elliptic functions. Here elementary arguments are used to verify the known first integral for a Somos-4 sequence and the two independent first integrals for a Somos-5 sequence, the latter being a result usually shown using the Weierstrass function for an associated elliptic curve.