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/''p'
nZ') 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 : \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. == See also ==