Prime power

In mathematics, a prime power is a positive integer power of a prime 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 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:

:$phi(p^n)\; =\; p^\{n-1\}\; phi(p)\; =\; p^\{n-1\}\; (p\; -\; 1)\; =\; p^n\; -\; p^\{n-1\}\; =\; p^n\; left(1\; -\; frac\{1\}\{p\}\; ight)$,

:$sigma\_0(p^n)\; =\; sum\_\{j=0\}^\{n\}\; p^\{0*j\}\; =\; sum\_\{j=0\}^\{n\}\; 1\; =\; n+1$,

:$sigma\_1(p^n)\; =\; sum\_\{j=0\}^\{n\}\; p^\{1*j\}\; =\; sum\_\{j=0\}^\{n\}\; p^\{j\}\; =\; frac\{p^\{n+1\}\; -\; 1\}\{p\; -\; 1\}$.

All prime powers are deficient numbers. A prime power "pⁿ" is an n-almost prime. It is not known whether a prime power "pⁿ" can be an amicable 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.