Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\beta$-continued fraction introduced by
Bernat. As a corollary of our theorem we show that for any quadratic Perron
number $\beta$, the $\beta$-continued fraction expansion of elements in
$\mathbb Q(\beta)$ is either finite of eventually periodic. The same holds for
$\beta$ being a square root of an integer. We also show that for certain 4
quadratic Perron numbers $\beta$, the $\beta$-continued fraction represents
finitely all elements of the quadratic field $\mathbb Q(\beta)$, 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.

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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg