We consider a one-parameter family of third order nonlinear recurrence relations. Each member of this family satisfies the singularity confinement test, has a conserved quantity, and moreover has the Laurent property: all of the iterates are Laurent polynomials in the initial data. However, we show that these recurrences are not Diophantine integrable according to the definition proposed by Halburd (2005 J. Phys. A: Math. Gen. 38 L1). Explicit bounds on the asymptotic growth of the heights of iterates are obtained for a special choice of initial data. As a by-product of our analysis, infinitely many solutions are found for a certain family of Diophantine equations, studied by Mordell, that includes Markoff’s equation. For some time there has been considerable interest in maps or discrete equations which are integrable. Various different criteria have been proposed as tests for integrability in the discrete setting. One of the earliest proposals was the singularity confinement test of Grammaticos, Ramani and Papageorgiou [12], which has proved to be an extremely useful tool for isolating discrete Painlevé equations (see e.g. [28]). However, after Hietarinta and Viallet’s discovery of some non-integrable equations with the singularity confinement property, they were led to introduce the zero algebraic entropy condition for integrability of rational maps [14], namely that the degree dn of the nth iterate of a map (as a rational function of the initial data) should satisfy limn→∞(log dn)/n = 0. The phenomenon of weak degree growth for integrable maps had been studied earlier by Veselov [33, 34], and the notion of algebraic entropy proposed in [14] is connected with other measures of growth such as Arnold complexity [1]. Since the work of Okamoto [22] it has been known that the Bäcklund transformations of differential Painlevé equations can be classified in terms of affine Weyl groups. Noumi and Yamada have shown that the symmetries of these ODEs are simultaneously compatible with associated discrete Painlevé equations [21], and this viewpoint has been explained geometrically by means of Cremona group actions on rational surfaces [30]. Discrete integrable systems are also related to the notions of discrete geometry [4] and discrete analytic functions

The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous family of sequences associated with an hyperelliptic curve of genus two defined by the affine model y 2 = 4x 5 + c4x 4 +... + c1x + c0. We show that the sequences associated with such curves satisfy bilinear recurrences of order 8. The proof requires an addition formula which involves the genus two Kleinian sigma function with its argument shifted by the Abelian image of the reduced divisor of a single point on the curve. The genus two recurrences are related to a Bäcklund transformation (BT) for an integrable Hamiltonian system, namely the discrete case (ii) Hénon-Heiles system. 1

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