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 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 $a = p_1^p_2^ \cdots p_k^$ and $b = q_1^q_2^ \cdots q_l^$ 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 $\frac=\frac$ 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

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Number theory