- Prime power
In
mathematics , a prime power is apositive integer power of aprime number .For example: 5=51, 9=32 and 16=24 are prime powers, while6, 15 and 36 are not. The twenty smallest prime powers are (sequence in
OEIS)::* 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, ...The prime powers are those positive integers that are divisible by just one prime number.
Properties
Algebraic properties
Every prime power (except powers of 2) has a primitive root; thus the multiplicative group of integers modulo pn (or equivalently, the
unit group of the ring mathbf{Z}/p^nmathbf{Z}) is cyclic.The number of elements of a
finite field is always a prime power and conversely, everyprime power occurs as the number of elements in some finite field (which is unique up toisomorphism ).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:
:phi(p^n) = p^{n-1} phi(p) = p^{n-1} (p - 1) = p^n - p^{n-1} = p^n left(1 - frac{1}{p} ight),
:sigma_0(p^n) = sum_{j=0}^{n} p^{0*j} = sum_{j=0}^{n} 1 = n+1,
:sigma_1(p^n) = sum_{j=0}^{n} p^{1*j} = sum_{j=0}^{n} p^{j} = frac{p^{n+1} - 1}{p - 1}.
All prime powers are
deficient number s. A prime power "pn" is an n-almost prime . It is not known whether a prime power "pn" can be anamicable number . If there is such a number, then "pn" must be greater than 101500 and "n" must be greater than 1400.ee also
*
Almost prime
*Semiprime References
*"Elementary Number Theory". Jones, Gareth A. and Jones, J. Mary. Springer-Verlag London Limited. 1998.
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