MULTIPLICATIVE DEPENDENCE OF TWO INTEGERS SHIFTED BY A ROOT OF UNITY.

In this note we prove a result on the multiplicative independence of the numbers m - α, n - α, where m > n are positive integers and α is a reciprocal algebraic number with the property that α+1/α has at least two real conjugates over ℚ lying in the interval (-∞, 2]. As an application, we show that for any positive integers m > n and k ⩾ 3 the numbers m - ζ_k, n - ζ_k, where ζ_k is the primitive kth root of unity, are multiplicatively independent except when (n, k) = (1, 6). This settles a recent conjecture of Madritsch and Ziegler. [ABSTRACT FROM AUTHOR]