In mathematics, a prime power is a positive integer that 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. 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, ... (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
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 pn (that is, the group of units of the ring Z/pnZ) 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
φ ( p n ) = p n − 1 φ ( p ) = p n − 1 ( p − 1 ) = p n − p n − 1 = p n ( 1 − 1 p ) , {\displaystyle \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),}
σ 0 ( p n ) = ∑ j = 0 n p 0 ⋅ j = ∑ j = 0 n 1 = n + 1 , {\displaystyle \sigma _{0}(p^{n})=\sum _{j=0}^{n}p^{0\cdot j}=\sum _{j=0}^{n}1=n+1,}
σ 1 ( p n ) = ∑ j = 0 n p 1 ⋅ j = ∑ j = 0 n p j = p n + 1 − 1 p − 1 . {\displaystyle \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}}.}
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.
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 doi:10.1007/978-1-4471-0613-5
