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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