Consecutive tuples of multiplicatively dependent integers

TL;DR

This paper investigates the existence and finiteness of consecutive pairs and triples of multiplicatively dependent integers, providing explicit classifications and effective methods for identifying such triples.

Contribution

It extends known results by proving finiteness and constructiveness of triples of multiplicatively dependent integers in consecutive positions, beyond pairs.

Findings

Only (2,8) and (3,9) are consecutive pairs of multiplicatively dependent integers > 1.

Finitely many triples of the form (a,b,c) with consecutive shifts are multiplicatively dependent.

Such triples can be explicitly determined.

Abstract

This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent integers larger than 1 are $(2,8)$ and $(3,9)$. For triples, we prove the following theorem: If $a\notin \{2,8\}$ is a fixed integer larger than 1, then there are only finitely many triples $(a,b,c)$ of pairwise distinct integers larger than 1 such that $(a,b,c)$, $(a+1,b+1,c+1)$ and $(a+2,b+2,c+2)$ are each multiplicatively dependent. Moreover, these triples can be determined effectively.