On quadratic fields generated by the Shanks sequence

Florian Luca, Igor E. Shparlinski
2009 Proceedings of the Edinburgh Mathematical Society  
Let u(n) = f (g n ), where g > 1 is integer and f (X) ∈ 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( u(n)) for n ∈ {M + 1, . . . , M + N }. Fields of this type include the Shanks fields and their generalizations.
doi:10.1017/s001309150700123x fatcat:g5fng5w4qjdztd4gjkxre5rkgm