In mathematics, a prime power is a positive integer that is a positive integer power of a single prime number.

For example: , and are prime powers, while

, and are not.

The sequence of prime powers begins:

<blockquote>2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, ... </blockquote>.

The prime powers are those positive integers that are divisible by exactly one prime number; in particular, the number 1 is not a prime power. Prime powers are also called primary numbers, as in the primary decomposition.

Properties

Algebraic properties

Prime powers are powers of prime numbers. Every prime power excluding powers of 2 greater than 4 has a primitive root; thus the multiplicative group of integers modulo p<sup>n</sup> (that is, the group of units of the ring Z/p<sup>n</sup>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<sub>0</sub>) and (σ<sub>1</sub>) of a prime power are calculated by the formulas

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

: <math>\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}.</math>

All prime powers are deficient numbers. A prime power p<sup>n</sup> is an n-almost prime. It is not known whether a prime power p<sup>n</sup> can be a member of an amicable pair. If there is such a number, then p<sup>n</sup> must be greater than 10<sup>1500</sup> and n must be greater than 1400.

See also

  • Almost prime
  • Fermi–Dirac prime
  • Perfect power
  • Semiprime

References

Further reading

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