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Finiteness and periodicity of continued fractions over quadratic number fields

We consider continued fractions with partial quotients in the ring of integers of a quadratic number field $K$ and we prove a generalization to such continued fractions of the classical theorem of Lagrange. A particular example of these continued fractions is the $β$-continued fraction introduced by Bernat. As a corollary of our theorem we show that for any quadratic Perron number $β$, the $β$-continued fraction expansion of elements in $\mathbb Q(β)$ is either finite of eventually periodic. The same holds for $β$ being a square root of an integer. We also show that for certain 4 quadratic Perron numbers $β$, the $β$-continued fraction represents finitely all elements of the quadratic field $\mathbb Q(β)$, thus answering questions of Rosen and Bernat. Based on the validity of a conjecture of Mercat, these are all quadratic Perron numbers with this feature.