# Prime power

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In mathematics, a prime power is a positive integer power of a single prime number. For example: 5 = 51, 9 = 32 and 16 = 24 are prime powers, while 6 = 2 × 3, 15 = 3 × 5 and 36 = 62 = 22 × 32 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 pn (or equivalently, the group of units of the ring Z/pnZ) 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 (σ0) and (σ1) of a prime power are calculated by the formulas:

${\displaystyle \phi (p^{n})=p^{n-1}\phi (p)=p^{n-1}(p-1)=p^{n}-p^{n-1}=p^{n}\left(1-{\frac {1}{p}}\right),}$
${\displaystyle \sigma _{0}(p^{n})=\sum _{j=0}^{n}p^{0\cdot j}=\sum _{j=0}^{n}1=n+1,}$
${\displaystyle \sigma _{1}(p^{n})=\sum _{j=0}^{n}p^{1\cdot j}=\sum _{j=0}^{n}p^{j}={\frac {p^{n+1}-1}{p-1}}.}$

All prime powers are deficient numbers. 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, prime powers play a key role, acting as indicators of lethal dangers in a maze-like cube structure.

## References

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