Units from square-roots of rational numbers

TL;DR

This paper explores the structure of units in quadratic integer rings derived from square roots of rational numbers, linking them to continued fraction convergents and revealing how to derive periodic continued fractions from rational approximations.

Contribution

It establishes a novel connection between units in quadratic rings and convergents of square roots of rational numbers, including a method to read periodic continued fractions from rational approximations.

Findings

Units >1 in certain quadratic rings relate to convergents of square roots.

Units of $ ext{Z}[\sqrt{DQ}]$ correspond to specific convergents before the end of each period.

The periodic continued fraction expansion can be derived from finite continued fractions of rational numbers.

Abstract

Let $D,Q$ be natural numbers, $(D,Q)=1$, such that $D/Q>1$ and $D/Q$ is not a square. Let $q$ be the smallest divisor of $Q$ such that $Q|\, q^2$. We show that the units $>1$ of the ring $\mathbb Z[\sqrt{Dq^2/Q}]$ are connected with certain convergents of $\sqrt{D/Q}$. Among these units, the units of $\mathbb Z[\sqrt{DQ}]$ play a special role, inasmuch as they correspond to the convergents of $\sqrt{D/Q}$ that occur just before the end of each period. We also show that the last-mentioned units allow reading the (periodic) continued fraction expansion of certain quadratic irrationals from the (finite) continued fraction expansion of certain rational numbers.