Ducci-sequences and cyclotomic polynomials
type: Item
instance of: scholarly article
title: Ducci-sequences and cyclotomic polynomials
zbMATH Open document ID: 1152.11005
author: Florian Breuer
author: Ernest Lötter
author: Brink van der Merwe
published in: Finite Fields and their Applications
publication date: 2008-09-25T00:00:00Z
review text: A Ducci sequence generated by \(X=(x_1,\ldots,x_n)\in {\mathbb Z}^n\) is the sequence \((X,DX,D^2X,\ldots)\), where \(DX=(| x_1-x_2| ,| x_2-x_3| ,\ldots,| x_n-x_1| )\). Every Ducci sequence gives rise to a cycle; i.e., there are integers \(i<j\) such that \(D^iX=D^jX\). When \(i\) and \(j\) are as small as possible, the Ducci sequence has period \(j-i\). It is known that for a given cycle all entries in all \(n\)-tuples are equal to either \(0\) or to a constant \(C\). Thus, one can study such periods by working with Ducci sequences in \({\mathbb F}_2^n\). In the paper under review, the authors determine the period of a given Ducci sequence in terms of the order of a certain polynomial related to the cyclotomic polynomials, as well as to the multiplicative orders of certain elements in finite fields of characteristic \(2\). They also compute several examples and study links between Ducci sequences, primitive polynomials and Artin's primtive root conjecture. The paper concludes with several interesting open problems.
reviewed by: Florian Luca
Mathematics Subject Classification ID: 11B50
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 5347269
zbMATH Keywords: Ducci sequences
zbMATH Keywords: cyclotomic polynomials
MaRDI profile type: Publication
full work available at URL: j.ffa.2005.11.003
OpenAlex ID: W1985140775
cites work: Circles of numbers
cites work: Q5695110
cites work: Q5754017
cites work: Q4842645
cites work: On Artin's conjecture.
cites work: Cycles of differences of integers
cites work: Q3491649
cites work: Q4936066
DOI: 10.1016/J.FFA.2005.11.003
description: scientific article; zbMATH DE number 5347269