On a family of generalized continued fraction expansions with periodic length going to infinity
type: Item
instance of: scholarly article
title: On a family of generalized continued fraction expansions with periodic length going to infinity
zbMATH Open document ID: 0837.11005
author: Josef Kühner
published in: Journal of Number Theory
publication date: 1996-05-20T00:00:00Z
review text: C. Levesque and G. Rhin exhibited a family of cubic fields \(\mathbb{Q} (\omega )\), \(\omega^3 - c^m \omega^2 - (c - 1) \omega - c^m\) \((c,m \in \mathbb{N}, c \geq 2)\), for which the Jacobi algorithm of \((\omega^2 - c^m \omega - c + 1, \omega)\) is purely periodic of length \(3m + 1\), thereby exhibiting an explicit unit through the Hasse-Bernstein formula. The author uses the Voronoï algorithm to show that this unit is fundamental in the order \(\mathbb{Z} (\omega)\) by writing down explicitly the period (again of length \(3m + 1)\) of this Voronoï algorithm. His results can be compared to those of \textit{Farhane-Dubois} (asymptotic results) [see Zbl 0837.11006 below] and those of \textit{B. Adam} (Thesis, Metz).
reviewed by: Claude Levesque
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R16
zbMATH DE Number: 792465
zbMATH Keywords: continued fractions
zbMATH Keywords: fundamental unit
zbMATH Keywords: cubic fields
zbMATH Keywords: Voronoï algorithm
MaRDI profile type: Publication
full work available at URL: jnth.1995.1075
OpenAlex ID: W2062711079
DOI: 10.1006/JNTH.1995.1075
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description: scientific article; zbMATH DE number 792465