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{{For|the electrical generator power rating|Prime power (electrical)}}

In [[mathematics]], a '''prime power''' is a [[positive integer]] [[exponentiation|power]] of a single [[prime number]].
For example: {{nowrap|1=5 = 51}}, {{nowrap|1=9 = 32}} and {{nowrap|1=16 = 24}} are prime powers, while
{{nowrap|1=6 = 2 × 3}}, {{nowrap|1=15 = 3 × 5}} and {{nowrap|1=36 = 62 = 22 × 32}} are not. The twenty smallest prime powers are:

:[[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[5 (number)|5]], [[7 (number)|7]], [[8 (number)|8]], [[9 (number)|9]], [[11 (number)|11]], [[13 (number)|13]], [[16 (number)|16]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[25 (number)|25]], [[27 (number)|27]], [[29 (number)|29]], [[31 (number)|31]], [[32 (number)|32]], [[37 (number)|37]], [[41 (number)|41]], ... {{OEIS|id=A000961}}.
The prime powers are those positive integers that are divisible by exactly one prime number; prime powers and related concepts are also called primary numbers, as in the [[primary decomposition]].

==Properties==
=== Algebraic properties ===
Prime powers are prime numbers and powers of prime numbers. Every prime power (except powers of 2) has a [[Primitive root modulo n|primitive root]]; thus the [[multiplicative group of integers modulo n|multiplicative group]] of integers modulo ''pn'' (or equivalently, the [[group of units]] of the [[ring (mathematics)|ring]] '''Z'''/''pn'''''Z''') is [[cyclic group|cyclic]].

The number of elements of a [[finite field]] is always a prime power and conversely, every
prime power occurs as the number of elements in some finite field (which is unique up to [[isomorphism]]).

=== Combinatorial properties ===
A property of prime powers used frequently in [[analytic number theory]] is that the set of prime powers which are not prime is a [[small set (combinatorics)|small set]] in the sense that the [[series (mathematics)|infinite sum]] of their reciprocals [[convergent series|converges]], although the primes are a large set.

=== Divisibility properties ===
The [[Euler's totient function|totient function]] (''φ'') and [[divisor function|sigma functions]] (''σ''0) and (''σ''1) of a prime power are calculated by the formulas:

:<math>\phi(p^n) = p^{n-1} \phi(p) = p^{n-1} (p - 1) = p^n - p^{n-1} = p^n \left(1 - \frac{1}{p}\right),</math>

:<math>\sigma_0(p^n) = \sum_{j=0}^{n} p^{0*j} = \sum_{j=0}^{n} 1 = n+1,</math>

:<math>\sigma_1(p^n) = \sum_{j=0}^{n} p^{1*j} = \sum_{j=0}^{n} p^{j} = \frac{p^{n+1} - 1}{p - 1}.</math>

All prime powers are [[deficient number]]s. A prime power ''pn'' is an ''n''-[[almost prime]]. It is not known whether a prime power ''pn'' can be an [[amicable number]]. If there is such a number, then ''pn'' must be greater than 101500 and ''n'' must be greater than 1400.

==Popular media==
In the 1997 film ''[[Cube (film)|Cube]]'', prime powers play a key role, acting as indicators of lethal dangers in a maze-like cube structure.

==See also==
* [[Perfect power]]
* [[Almost prime]]
* [[Semiprime]]

==References==
*''Elementary Number Theory''. Jones, Gareth A. and Jones, J. Mary. Springer-Verlag London Limited. 1998.

{{Classes of natural numbers}}
[[Category:Prime numbers]]
[[Category:Exponentials]]

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