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We consider a one-parameter family of third order nonlinear recurrencerelations. Each member of this family satisfies the singularity confinementtest, has a conserved quantity, and moreover has the Laurent property: all ofthe iterates are Laurent polynomials in the initial data. However, we show thatthese recurrences are not Diophantine integrable according to the definitionproposed by Halburd. Explicit bounds on the asymptotic growth of the heights ofiterates are obtained for a special choice of initial data. As a by-product ofour analysis, infinitely many solutions are found for a certain family ofDiophantine equations, studied by Mordell, that includes Markoff's equation.Comment: 7 pages; statement of Proposition 2 and minor typos corrected. To  appear in J. Phys. A: Math. Ge

Diophantine non-integrability of a third-order recurrence with the Laurent property