power of an integer

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


where rad(n) is the radicalPlanetmathPlanetmathPlanetmathPlanetmath of the integer n.11Since |n|1, we have rad(n)1 also, so the denominator will not be equal to zero

If n=mk, then P(n)=kP(m); in particular, if n is a prime power, n=pk, 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,


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