On the number of distinct quadratic fields generated by the Shanks sequence

TL;DR

This paper investigates the distribution of quadratic fields generated by sequences of the form u(n)=f(g^n), improving bounds on how often these fields coincide with a fixed quadratic field, using advanced sieve and character sum techniques.

Contribution

It provides new bounds on the frequency of quadratic fields generated by sequences related to Shanks fields, enhancing previous results with novel analytic methods.

Findings

Improved upper bounds on the number of n with Q(√u(n)) equal to a fixed quadratic field.

Application of the square sieve and new bounds on character sums.

Generalization of results to sequences including Shanks fields.

Abstract

Let $g>1$ be an integer and $f(X)\in{\mathbb Z}[X]$ a polynomial of positive degree with no multiple roots, and put $u(n)=f(g^n)$. In this note, we study the sequence of quadratic fields ${\mathbb Q}(\sqrt{u(n)}\,)$ as $n$ varies over the consecutive integers $M+1,\ldots,M+N$. Fields of this type include Shanks fields and their generalizations. Using the square sieve together with new bounds on character sums, we improve an upper bound of Luca and Shparlinski (2009) on the number of $n \in \{M+1,\ldots,M+N\}$ with ${\mathbb Q}(\sqrt{u(n)}\,) = {\mathbb Q}(\sqrt{s}\,)$ for a given squarefree integer $s$.