Prime power
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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, 15 and 36 are not. The twenty smallest prime powers are (sequence A000961 in OEIS):
The prime powers are those positive integers that are divisible by just one prime number.
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[edit] Properties
[edit] 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 
) 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).
[edit] 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.
[edit] Divisibility properties
The totient function (φ) and sigma functions (σ0) and (σ1) of a prime power are calculated by the formulas:
,
,
.
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.
[edit] In 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.
[edit] See also
[edit] References
- Elementary Number Theory. Jones, Gareth A. and Jones, J. Mary. Springer-Verlag London Limited. 1998.
