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, $a^{n}=b^{m}$ 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 6 and 12 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.

References

[2]

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

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[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.