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, $a^n=b^m$ implies $n=m=0$. Two integers which are not multiplicatively independent are said to be multiplicatively dependent.

As examples, 36 and 216 are multiplicatively dependent since $36^3=(6^2{)}^3=(6^3{)}^2=216^2$, 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 $\log (a)/\log (b)$ is irrational. This property holds independently of the base of the logarithm.

Let $a=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_k^{{\alpha }_k}$ and $b=q_1^{{\beta }_1}{q}_2^{{\beta }_2}\cdots q_l^{{\beta }_l}$ be the canonical representations of a and b. The integers a and b are multiplicatively dependent if and only if k = l, $p_i=q_i$ and $\frac{{\alpha }_i}{{\beta }_i}=\frac{{\alpha }_j}{{\beta }_j}$ 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 $a^n=b^m$. 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.