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- W4387307430 abstract "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." @default.
- W4387307430 created "2023-10-04" @default.
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- W4387307430 date "2019-11-18" @default.
- W4387307430 modified "2023-10-05" @default.
- W4387307430 title "Finiteness and periodicity of continued fractions over quadratic number fields" @default.
- W4387307430 doi "https://doi.org/10.48550/arxiv.1911.07670" @default.
- W4387307430 hasPublicationYear "2019" @default.
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