Multiplicative independence

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, two positive <a href="/facts/Integer/PUwwrawx">integers</a> 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
 
 
 
 
 {\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.

Multiplicative independence open-in-new

Multiplicative independence