Remarks and problems on finite and periodic continued fractions
type: Item
instance of: scholarly article
title: Remarks and problems on finite and periodic continued fractions
zbMATH Open document ID: 0808.11007
author: Michel Mendès France
published in: L'Enseignement Mathématique. 2e Série
publication date: 1995-03-09T00:00:00Z
review text: The author raises eight interesting problems with varying difficulty. The first problem concerns properties of Pisot numbers. Two further problems are as follows: (1) Let \(\delta(x)\) be the length of the continued fraction expansion (c.f.e.) of the rational \(x\). Is it true that for all integers \(a\) and \(b\), \(1<a<b\), \((a,b)=1\), we have \[ \lim_{n\to\infty} 1/n\;\delta ((a/b)^ n)= 12/\pi^ 2 \ln 2\ln b\;? \] (2) Let \(x\) be a real quadratic number and let \(\pi(x)\) be the period of the c.f.e. of \(x\). Is it true that \(\sup_ n \pi(x^ n)= \infty\)? This last problem has now been partially solved.
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 597692
zbMATH Keywords: Möbius maps
zbMATH Keywords: Pisot numbers
zbMATH Keywords: length
zbMATH Keywords: continued fraction expansion
zbMATH Keywords: period
reviewed by: Tonko Tonkov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 597692