Diophantine representations of linear recurrent sequences.~II

M. A. Vsemirnov

Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part X, Tome 241 (1997), pp. 5-29

Résumé

Direct constructions of diophantine representations of linear recurrent sequences are considered. Diophantine representations of the sets of values of third-order sequences with negative discriminant are found. As an auxiliary problem we study the structure of the multiplicative group of the ring $\mathbb Z[\lambda]$, where $\lambda$ is an invertible algebraic number (unit) in a real quadratic field or in a cubic field of a negative discriminant. Tge index of the subgroup $\langle\pm\lambda^n\mid n\in\mathbf Z\rangle$ in the group $(\mathbf Z[\lambda])^*$ and the generator of $(\mathbf Z[\lambda])^*$ are evaluated explicitly.