Two families of periodic Jacobi algorithms with period lengths going to infinity
type: Item
instance of: scholarly article
title: Two families of periodic Jacobi algorithms with period lengths going to infinity
zbMATH Open document ID: 0723.11032
DOI: 10.1016/S0022-314X(05)80035-6
author: Claude Levesque
author: Georges Rhin
published in: Journal of Number Theory
publication date: 1991-01-01T00:00:00Z
review text: Let \(c\geq 2\) and \(m\geq 1\) be fixed integers. The authors establish that the Jacobi-Perron expansion of the vector \((x^ 2-(c^ m+c-1)x,x)\) is purely periodic of period length \(4m+1\), where x is selected to satisfy \(c^ m+c-1<x<c^ m+c.\) The key to their solution is to show that \(g(x)=x^ 3-(c^ m+c-1)x^ 2-(c^ m-1)x-c^ m\) has only one real root and that root x satisfies the previous inequalities. By a similar consideration it is also established that the Jacobi-Perron expansion of the vector \((x^ 2-c^ mx,x)\) is purely periodic, this time with period length \(3m+1\).
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11A67
zbMATH DE Number: 4191793
zbMATH Keywords: Jacobi-Perron expansion
zbMATH Keywords: period length
reviewed by: Janos Galambos
MaRDI profile type: Publication
cites work: The Jacobi-Perron algorithm its theory and application
cites work: Q5607312
cites work: Q4050688
cites work: Q4050687
cites work: A class of fundamental units and some classes of Jacobi-Perron algorithms in pure cubic fields
cites work: Q5661585
full work available at URL: s0022-314x(05)80035-6
OpenAlex ID: W2080914225
description: scientific article; zbMATH DE number 4191793