Prime power

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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, while 6, 15 and 36 are not. The twenty smallest prime powers are (sequence A000961 in OEIS):

The prime powers are those positive integers that are divisible by just one prime number.

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[edit] Properties

[edit] Algebraic properties

Every prime power has a primitive root; thus the multiplicative group of integers modulo pn (or equivalently, the unit group of the ring \mathbf{Z}/p^n\mathbf{Z}) 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).

[edit] 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.

[edit] 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}\right),
\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.

[edit] See also

[edit] References

  • Elementary Number Theory. Jones, Gareth A. and Jones, J. Mary. Springer-Verlag London Limited. 1998.