# Counting fundamental solutions to the Pell equation with prescribed size

## TL;DR

This paper investigates the number of fundamental solutions to Pell equations with size constraints, providing new lower bounds by applying advanced exponential sum techniques, thus advancing understanding of Pell solutions' distribution.

## Contribution

It introduces a novel application of the $q$-analogue of van der Corput method to algebraic exponential sums, improving bounds on Pell solutions compared to previous work.

## Key findings

- Established new lower bounds for the count of Pell solutions
- Applied the $q$-analogue of van der Corput method to algebraic sums
- Enhanced previous results by Fouvry on Pell equation solutions

## Abstract

The cardinality of the set of $D\leqslant x$ for which the fundamental solution of the Pell equation $t^2-Du^2=1$ is less than $D^{\frac{1}{2}+\alpha}$ with $\alpha\in[\frac{1}{2},1]$ is studied and certain lower bounds are obtained, improving previous results of Fouvry by introducing the $q$-analogue of van der Corput method to algebraic exponential sums with smooth moduli.