Prime power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: 7 = 71, 9 = 32 and 64 = 26 are prime powers, while 6 = 2 × 3, 12 = 22 × 3 and 36 = 62 = 22 × 32 are not.
For the electrical generator power rating, see Prime power (electrical).
The sequence of prime powers begins:
(sequence A246655 in the OEIS).
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[edit]
Algebraic properties[edit]
Prime powers are powers of prime numbers. Every prime power (except powers of 2 greater than 4) has a primitive root; thus the multiplicative group of integers modulo pn (that is, the group of units of the ring Z/pnZ) is cyclic.[1]
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).[2]
Combinatorial properties[edit]
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.[3]
Divisibility properties[edit]
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 a member of an amicable pair. If there is such a number, then pn must be greater than 101500 and n must be greater than 1400.