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arXiv · 2211.12160

Units from square-roots of rational numbers

Kurt Girstmair
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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.

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Kurt Girstmair. 2022-11-22. Units from square-roots of rational numbers. https://arxiv.org/abs/2211.12160

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