On quadratic fields generated by the Shanks sequence
Florian Luca, Igor E. Shparlinski · Proceedings of the Edinburgh Mathematical Society · 2009
Abstract Let u(n)=f(gn), where g > 1 is integer and f(X) ∈ ℤ[X] is non-constant and has no multiple roots. We use the theory of $\mathcal{S}$ -unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among $\mathbb{Q}(\sqrt{u(n)})$ for n ∈ $\{M+1,\dots,M+N\}$ . Fields of this type include the Shanks fields and their generalizations.