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Continued fractions in 2-stage Euclidean quadratic fields. (English) Zbl 1263.11095

A number field \(K\) is called Euclidean (with respect to the norm) if for every nonzero pair \(\alpha, \beta \in {\mathcal O}_K\) there exists a division chain \(\alpha = \beta \gamma_1 + \delta_1\), \(\beta = \delta_1 \gamma_1 + \delta_2\), \(\ldots\) in which the absolute values of the norms of the remainders decrease (\(|N(\delta_1)| > |N(\delta_2)| > \ldots \)) until the remainder becomes \(0\). We say that \(K\) is two-stage Euclidean if the norms decrease at least in every other step; thus we have \(|N(\delta_1)| > |N(\delta_2)|\) or \(|N(\delta_1)| > |N(\delta_3)|\) etc.
In this article, the authors extend the list of known examples of \(2\)-stage Euclidean quadratic number fields by showing that all real quadratic number fields with class number \(1\) and discriminant below \(8000\) are \(2\)-stage Euclidean.

MSC:

11R11 Quadratic extensions
11A55 Continued fractions
13F07 Euclidean rings and generalizations
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References:

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