In number theory, two positive integers a and b are said to be multiplicatively independent[1] if their only common integer power is 1.
implies
Two integers which are not multiplicatively independent are said to be multiplicatively dependent.
As examples, 36 and 216 are multiplicatively dependent since
, whereas 2 and 3 are multiplicatively independent.
Being multiplicatively independent admits some other characterizations.
a and b are multiplicatively independent if and only if
is irrational.
This property holds independently of the base of the logarithm.
α
α
α
β
β
β
be the canonical representations of a and b.
The integers a and b are multiplicatively dependent if and only if k = l,
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
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.