Towards a theorem of Cobham for Gaussian integers
type: Item
instance of: scholarly article
title: Towards a theorem of Cobham for Gaussian integers
zbMATH Open document ID: 1071.68089
author: Georges Hansel
author: Taoufik Safer
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2005-06-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1074791328
review text: A celebrated theorem of \textit{A. Cobham} asserts that if a set of integers is both \(p\)- and \(q\)-recognizable, where \(p\) and \(q\) are two integers \(\geq 2\) that are multiplicatively independent, then this set is the union of a finite set and of finitely many arithmetic progressions [Math. Syst. Theory 3, 186--192 (1969; Zbl 0179.02501)]. The authors give a first step for a similar result where \(p\) and \(q\) are replaced by \(-p+i\) and \(-q+i\) \((p>0\), \(q>0\), \(p\neq q\), \(p\) and \(q\) integers) and the set of integers is replaced by a set of Gauss integers: they prove that, assuming the four exponentials conjecture, such a set is syndetic (i.e., without ``big holes''). Please note that Reference [11] appeared with a slightly different title [\textit{P. Mihăilescu}, ``Primary cyclotomic units and a proof of Catalan's conjecture'', J. Reine Angew. Math. 572, 167--195 (2004; Zbl 1067.11017)].
reviewed by: Jean-Paul Allouche
Mathematics Subject Classification ID: 68R15
Mathematics Subject Classification ID: 11B85
Mathematics Subject Classification ID: 68Q45
zbMATH DE Number: 2174591
zbMATH Keywords: Cobham theorem
zbMATH Keywords: finite automata
zbMATH Keywords: \(p\)-recognizability
zbMATH Keywords: complex bases
zbMATH Keywords: four exponentials conjecture
MaRDI profile type: Publication
Recommended article: Q4815661
Recommended article: Q5864866
Recommended article: Q3106230
Recommended article: The Prouhet-Tarry-Escott problem for Gaussian integers
Recommended article: Q3106116
Recommended article: Q4347183
Recommended article: Q5317240
Recommended article: Fermat's Last Theorem for Gaussian Integer Exponents
Recommended article: On the set of divisors of Gaussian integers
Recommended article: A structure theorem for multiplicative functions over the Gaussian integers and applications
description: scientific article; zbMATH DE number 2174591