Revista Matemática Iberoamericana

Volume 34, Issue 4, 2018, pp. 1789–1808
DOI: 10.4171/rmi/1043

Published online: 2018-12-03

Multiplicative dependence of the translations of algebraic numbers

Artūras Dubickas[1] and Min Sha[2]

(1) Vilnius University, Lithuania
(2) Macquarie University, Sydney, Australia

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 $10^{10}$.

Keywords: Multiplicative dependence, multiplicative independence, Pillai’s equation, ABC conjecture

Dubickas Artūras, Sha Min: Multiplicative dependence of the translations of algebraic numbers. Rev. Mat. Iberoam. 34 (2018), 1789-1808. doi: 10.4171/rmi/1043