# Some more Long Continued Fractions, I

## TL;DR

This paper constructs infinite families of polynomials whose square roots have long, controlled-period continued fraction expansions and provides methods to compute fundamental units in related quadratic fields.

## Contribution

It introduces a systematic way to generate polynomials with long continued fraction periods and efficient techniques for fundamental unit computation in quadratic fields.

## Key findings

- Constructed families of polynomials with arbitrarily long continued fraction periods.
- Developed methods for quick computation of fundamental units.
- Demonstrated control over period length via parameter k.

## Abstract

In this paper we show how to construct several infinite families of polynomials $D(\bar{x},k)$, such that $\sqrt{D(\bar{x},k)}$ has a regular continued fraction expansion with arbitrarily long period, the length of this period being controlled by the positive integer parameter $k$. We also describe how to quickly compute the fundamental units in the corresponding real quadratic fields.