Prime power

For the electrical generator power rating, see Prime power (electrical).

In mathematics, a prime power is a positive integer power of a 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 A000961 in OEIS)

The prime powers are those positive integers that are divisible by just one prime number; prime powers and related concepts are also called primary numbers, as in the primary decomposition.

Properties

Algebraic properties

Every prime power (except powers of 2) has a primitive root; thus the multiplicative group of integers modulo pⁿ (or equivalently, the unit group of the ring $\mathbf{Z}/p^n\mathbf{Z}$) 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 (σ₀) and (σ₁) of a prime power are calculated by the formulas:

$\phi(p^n) = p^{n-1} \phi(p) = p^{n-1} (p - 1) = p^n - p^{n-1} = p^n \left(1 - \frac{1}{p}\right)$,

$\sigma_0(p^n) = \sum_{j=0}^{n} p^{0*j} = \sum_{j=0}^{n} 1 = n+1$,

$\sigma_1(p^n) = \sum_{j=0}^{n} p^{1*j} = \sum_{j=0}^{n} p^{j} = \frac{p^{n+1} - 1}{p - 1}$.

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.

See also

• Almost prime
• Semiprime

References

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