# Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields

## TL;DR

This paper constructs polynomial solutions to Pell's equation linked to fundamental units in real quadratic fields, providing explicit formulas and continued fraction expansions, thus advancing methods for algebraic number theory analysis.

## Contribution

It introduces a systematic way to generate polynomial solutions to Pell's equation associated with minimal solutions, enhancing understanding of fundamental units in real quadratic fields.

## Key findings

- Polynomial solutions $(c(t),h(t),f(t))$ are derived for given minimal solutions.
- Continued fraction expansions of $\
- f(t)\

## Abstract

Finding polynomial solutions to Pell's equation is of interest as such solutions sometimes allow the fundamental units to be determined in an infinite class of real quadratic fields. In this paper, for each triple of positive integers $(c,h,f)$ satisfying \[c^{2}-f\,h^{2}=1, \] where $(c,h)$ are the smallest pair of integers satisfying this equation, several sets of polynomials $(c(t),h(t),f(t))$ which satisfy \[c(t)^{2}-f(t)\,h(t)^{2}=1 \text{ and } (c(0),h(0),f(0)) = (c,h,f) \] are derived. Moreover, it is shown that the pair $(c(t),h(t))$ constitute the fundamental polynomial solution to the Pell's equation above. The continued fraction expansion of $\sqrt{f(t)}$ is given in certain general cases (for example, when the continued fraction expansion of $\sqrt{f}$ has odd period length, or even period length or has period length $\equiv 2 \mod{4}$ and the middle quotient has a particular form etc). Some applications to determining the fundamental unit in real quadratic fields is also discussed.