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) 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]^{[better source needed]}

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]^{[better source needed]}

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

^ Daileda, Ryan C. (6 April 2018). "The Structure of (Z/nZ)×" (PDF). Trinity University Mathematics. Retrieved 25 December 2022.
^ Conrad, Keith. "Finite fields" (PDF). Kconrad.math.uconn.edu. Retrieved 25 December 2022.
^ 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 – via JSTOR.