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 all integer n and m,
a
n
=
b
m
implies
n
=
m
=
0
. Two integers which are not multiplicatively independent are said to be multiplicatively dependent.
For example, 36 and 216 are multiplicatively dependent since
36
3
=
(
6
2
)
3
=
(
6
3
)
2
=
216
2
and 6 and 12 are multiplicatively independent
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
α
1
p
2
α
2
⋯
p
k
α
k
and
b
=
q
1
β
1
q
2
β
2
⋯
q
l
β
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
α
i
β
i
=
α
j
β
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
. 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.
