Multiplicative dependence of the translations of algebraic numbers

Abstract

In this paper, we first prove that given pairwise distinct algebraic numbers \alpha_1, \ldots, \alpha_n , the numbers \alpha_1+t, \ldots, \alpha_n+t are multiplicatively independent for all sufficiently large integers t . Then, for a pair (a,b) of distinct integers, we study how many pairs (a+t,b+t) are multiplicatively dependent when t runs through the set integers \mathbb Z . Assuming the ABC conjecture we show that there exists a constant C_1 such that for any pair (a,b)\in \mathbb Z^2 , a \ne b , there are at most C_1 values of t \in \mathbb Z such that (a+t,b+t) are multiplicatively dependent. For a pair (a,b) \in \mathbb Z^2 with difference b-a=30 we show that there are 13 values of t \in \mathbb Z for which the pair (a+t,b+t) is multiplicatively dependent. We further conjecture that 13 is the largest number of such translations for any such pair (a,b) and prove this for all pairs (a,b) with difference at most 10^{10} .