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arXiv · 2209.04886

On the equivalence of certain quadratic irrationals

Kurt Girstmair
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Abstract

This paper deals with quadratic irrationals of the form $m/q+\sqrt v$ for fixed positive integers $v$ and $q$, $v$ not a square, and varying integers $m$, $(m,q)=1$. Two numbers $m/q+\sqrt v$, $n/q+\sqrt v$ of this kind are equivalent (in a classical sense) if their continued fraction expansions can be written with the same period. We give a necessary and sufficient condition for the equivalence in terms of solutions of Pell's equation. Moreover, we determine the number of equivalence classes to which these quadratic irrationals belong.

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Kurt Girstmair. 2022-09-11. On the equivalence of certain quadratic irrationals. https://arxiv.org/abs/2209.04886

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