Type: Article

On quadratic fields generated by the Shanks sequence

Florian Luca; Igor E. Shparlinski;

Abstract:

Let u(n) = f(g(n)), where g > 1 is integer and f(X) is an element of Z[X] is non-constant and has no multiple roots. We use the theory of S-unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among Q(root u(n)) for n is an element of {M + 1, ..., M + N}. Fields of this type include the Shanks fields and their generalizations.