Prime power

In mathematics, a prime power 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 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, ... .

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 (except powers of 2) has a primitive root; thus the multiplicative group of integers modulo ''p''^''n'' (i.e. 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:
:$\varphi(p^n) = p^ \varphi(p) = p^ (p - 1) = p^n - p^ = p^n \left(1 - \frac\right),$
:$\sigma_0(p^n) = \sum_^ p^ = \sum_^ 1 = n+1,$
:$\sigma_1(p^n) = \sum_^ p^ = \sum_^ p^ = \frac.$

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.

See also

*Almost prime
*Perfect power
*Semiprime

References

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

{{Classes of natural numbers
Prime numbers
Exponentials