power of an integer

Let $n$ be a non zero integer of absolute value not equal to one. The power of $n$, written $P(n)$ is defined by :

$$P(n)=\frac{\log |n|}{\log \mathrm{rad}(n)}.$$

where $\mathrm{rad}(n)$ is the radical of the integer $n$.

If $n=m^k$, then $P(n)=kP(m)$; in particular, if $n$ is a prime power, $n=p^k$, then $P(n)=k$. This observation explains why the term “power” is used for this concept. At the same time, it is worth pointing out that, in general, the power of an integer will not itself be an integer. For instance,

$$P(12)=\frac{\log 12}{\log \mathrm{rad}(12)}=\frac{\log 12}{\log 6}=1.3868\mathrm{\dots }$$

Note that it doesn’t matter what base one uses to compute the logarithm (as long as one uses the same base to compute the logarithm on the numerator and in the denominator!) because, upon changing base, both numerator and denominator will be multiplied by the same factor.

Title	power of an integer
Canonical name	PowerOfAnInteger
Date of creation	2013-03-22 14:22:17
Last modified on	2013-03-22 14:22:17
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	18
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 11N25
Synonym	power
Related topic	RadicalOfAnInteger