More on consecutive multiplicatively dependent triples of integers

TL;DR

This paper classifies triples of integers with specific multiplicative dependence properties, extending previous work and employing elementary and advanced Diophantine approximation techniques.

Contribution

It provides a complete classification of such triples when the first element is 2, advancing understanding of multiplicative dependence in integer triples.

Findings

Complete classification of triples with a=2

Extension of previous work on multiplicative dependence

Use of Diophantine approximation methods

Abstract

In this paper, we extend recent work of the third author and Ziegler on triples of integers $(a,b,c)$, with the property that each of $(a,b,c)$, $(a+1,b+1,c+1)$ and $(a+2,b+2,c+2)$ is multiplicatively dependent, completely classifying such triples in case $a=2$. Our techniques include a variety of elementary arguments together with more involved machinery from Diophantine approximation.