Multiplicative independence

In number theory, two positive integers a and b are said to be multiplicatively independent if their only common integer power is 1. That is, for integers n and m, [math]\displaystyle{ a^n=b^m }[/math] implies [math]\displaystyle{ n=m=0 }[/math]. Two integers which are not multiplicatively independent are said to be multiplicatively dependent.

As examples, 36 and 216 are multiplicatively dependent since [math]\displaystyle{ 36^3=(6^2)^3=(6^3)^2=216^2 }[/math], whereas 2 and 3 are multiplicatively independent.

Properties

Being multiplicatively independent admits some other characterizations. a and b are multiplicatively independent if and only if [math]\displaystyle{ \log(a)/\log(b) }[/math] is irrational. This property holds independently of the base of the logarithm.

Let [math]\displaystyle{ a = p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_k^{\alpha_k} }[/math] and [math]\displaystyle{ b = q_1^{\beta_1}q_2^{\beta_2} \cdots q_l^{\beta_l} }[/math] be the canonical representations of a and b. The integers a and b are multiplicatively dependent if and only if k = l, [math]\displaystyle{ p_i=q_i }[/math] and [math]\displaystyle{ \frac{\alpha_i}{\beta_i}=\frac{\alpha_j}{\beta_j} }[/math] for all i and j.

Applications

Büchi arithmetic in base a and b define the same sets if and only if a and b are multiplicatively dependent.

Let a and b be multiplicatively dependent integers, that is, there exists n,m>1 such that [math]\displaystyle{ a^n=b^m }[/math]. The integers c such that the length of its expansion in base a is at most m are exactly the integers such that the length of their expansion in base b is at most n. It implies that computing the base b expansion of a number, given its base a expansion, can be done by transforming consecutive sequences of m base a digits into consecutive sequence of n base b digits.