Multiplicative independence

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Multiplicative independence" – news · newspapers · books · scholar · JSTOR (April 2016)

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 {\displaystyle a^{n}=b^{m}} implies n = m = 0 {\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 36 3 = ( 6 2 ) 3 = ( 6 3 ) 2 = 216 2 {\displaystyle 36^{3}=(6^{2})^{3}=(6^{3})^{2}=216^{2}} , whereas 2 and 3 are multiplicatively independent.

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

Let a = p 1 α 1 p 2 α 2 ⋯ p k α k {\displaystyle a=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}} and b = q 1 β 1 q 2 β 2 ⋯ q l β l {\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, p i = q i {\displaystyle p_{i}=q_{i}} and α i β i = α j β j {\displaystyle {\frac {\alpha _{i}}{\beta _{i}}}={\frac {\alpha _{j}}{\beta _{j}}}} for all i and j.

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 {\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.

^[2]