Consecutive tuples of multiplicatively dependent integers

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∉{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.