# Prime power

{{#invoke:Hatnote|hatnote}}

In mathematics, a **prime power** is a positive integer power of a single prime number.
For example: 5 = 5^{1}, 9 = 3^{2} and 16 = 2^{4} are prime powers, while
6 = 2 × 3, 15 = 3 × 5 and 36 = 6^{2} = 2^{2} × 3^{2} are not. The twenty smallest prime powers are:

- 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, ... (sequence A246655 in OEIS).

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 powers of prime numbers. Every prime power (except powers of 2) has a primitive root; thus the multiplicative group of integers modulo *p ^{n}* (or equivalently, the group of units of the ring

**Z**/

*p*

^{n}**Z**) is 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 in the sense that the infinite sum of their reciprocals converges, although the primes are a large set.

### Divisibility properties

The totient function (*φ*) and sigma functions (*σ*_{0}) and (*σ*_{1}) of a prime power are calculated by the formulas:

All prime powers are deficient numbers. A prime power *p ^{n}* is an

*n*-almost prime. It is not known whether a prime power

*p*can be an amicable number. If there is such a number, then

^{n}*p*must be greater than 10

^{n}^{1500}and

*n*must be greater than 1400.

## Popular media

In the 1997 film *Cube*, prime powers play a key role, acting as indicators of lethal dangers in a maze-like cube structure.

## See also

## References

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