Prime power

In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: $7 = 7 1$ , $9 = 3 2$ and $64 = 2 6$ are prime powers, while $6 = 2 \times 3$ , $12 = 2 2 \times 3$ and $36 = 6 2 = 2 2 \times 3 2$ 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 (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

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

The totient function (φ) and sigma functions (σ₀) and (σ₁) of a prime power are calculated by the formulas

\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),

\sigma _{0}(p^{n})=\sum _{j=0}^{n}p^{0\cdot j}=\sum _{j=0}^{n}1=n+1,

\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.

References

^ Crandall, Richard; Pomerance, Carl B. (2005). Prime Numbers: A Computational Perspective (2nd ed.). Springer. p. 40. ISBN 9780387289793.
^ Koblitz, Neal (2012). A Course in Number Theory and Cryptography. Graduate Texts in Mathematics. Vol. 114. Springer. p. 34. ISBN 9781468403107.
^ Bayless, Jonathan; Klyve, Dominic (November 2013). "Reciprocal Sums as a Knowledge Metric: Theory, Computation, and Perfect Numbers". The American Mathematical Monthly. 120 (9): 822–831. doi:10.4169/amer.math.monthly.120.09.822. JSTOR 10.4169/amer.math.monthly.120.09.822. S2CID 12825183 – via JSTOR.