Neha Patil (Editor)

Multiplicative independence

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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.

Contents

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

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 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.

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

Multiplicative independence Wikipedia


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