Arithmetic Progressions in a Unique Factorization Domain

TL;DR

This paper extends Pillai's result on consecutive integers to arithmetic progressions in unique factorization domains of characteristic zero, providing a new proof and broader applicability.

Contribution

It offers a new proof of a generalized Pillai's theorem for arithmetic progressions in UFDs of characteristic zero, expanding the scope of previous results.

Findings

Generalization of Pillai's theorem to UFDs

New proof technique for arithmetic progressions

Extension to characteristic zero domains

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.