Multiplicative independence

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

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

Let ${\displaystyle a=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}}$ and ${\displaystyle b=q_{1}^{\beta _{1}}q_{2}^{\beta _{2}}\cdots q_{l}^{\beta _{l}}}$ be the canonical representations of a and b. The integers a and b are multiplicatively dependent if and only if k = l, ${\displaystyle p_{i}=q_{i}}$ and ${\displaystyle {\frac {\alpha _{i}}{\beta _{i}}}={\frac {\alpha _{j}}{\beta _{j}}}}$ 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 ${\displaystyle a^{n}=b^{m}}$ . 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.

References

^[2]

1. Bès, Alexis. "A survey of Arithmetical Definability". Archived from the original on 28 November 2012. Retrieved 27 June 2012.
2. Bruyère, Véronique; Hansel, Georges; Michaux, Christian; Villemaire, Roger (1994). "Logic and p-recognizable sets of integers" (PDF). Bull. Belg. Math. Soc. 1: 191--238.