Arithmetic progressions in a unique factorization domain

Arithmetic progressions in a unique factorization domain


Sudhir R. Ghorpade

Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India

E-Mail: srg@math.iitb.ac.in

and

Samrith Ram

Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India

E-Mail: samrith@gmail.com


Abstract

Pillai showed that any sequence of consecutive integers with at most 16 terms possesses one term that is relatively prime to all the others. We give a new proof of a slight generalization of this result to arithmetic progressions of integers and further extend it to arithmetic progressions in unique factorization domains of characteristic zero.


1 Introduction 1
2 Arithmetic progressions of integers 2
3 GCD domains and decomposition numbers 6
4 Arithmetic progressions in GCD domains 8
References 11


This paper is published in Acta Arithmetica, Vol. 54, No. 2 (2012), pp. 161-171. [Preprint Version: arXiv.math/1108.1267v1 (August 2011)]

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