Continued Fractions of Quadratic Numbers

TL;DR

This paper reviews known results on continued fractions, focusing on quadratic numbers, especially square roots, highlighting both classical and potentially new findings about their periodic continued fraction expansions.

Contribution

It summarizes existing knowledge and presents new results on the continued fraction representations of quadratic numbers, particularly square roots of natural numbers.

Findings

Descriptions of periods of continued fractions for 

Identification of new results on continued fractions of quadratic numbers

Compilation of classical and recent findings

Abstract

In this paper, we will first summarize known results concerning continued fractions. Then we will limit our consideration to continued fractions of quadratic numbers. The second author described periods and sometimes precise form of continued fractions of $\sqrt{N}$, where $N$ is a natural number. In cases where we were able to find such results in literature, we recall the original authors, however many results seem to be new.