Prime power

 ( Prime Numbers ) 

In mathematics, a prime power is a positive integer which is a positive integer exponentiation of a single prime number. For example: , and are prime powers, while , and 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, …

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The prime powers are those positive integers that are divisor 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.

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Properties

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Algebraic properties Prime powers are powers of prime numbers. Every prime power (except powers of 2 greater than 4) has a primitive root; thus the multiplicative group of integers modulo p ⁿ (that is, the group of units of the ring Z/ p ⁿ Z) is Cyclic group.

(2025). 9780387289793, Springer. . ISBN 9780387289793

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).

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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.

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Divisibility properties The totient function ( φ) and divisor function ( σ₀) and ( σ₁) of a prime power are calculated by the formulas

$\backslash varphi(p^n)\; =\; p^\{n-1\}\; \backslash varphi(p)\; =\; p^\{n-1\}\; (p\; -\; 1)\; =\; p^n\; -\; p^\{n-1\}\; =\; p^n\; \backslash left(1\; -\; \backslash frac\{1\}\{p\}\backslash right),$

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

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

All prime powers are . 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 numbers. If there is such a number, then pⁿ must be greater than 10¹⁵⁰⁰ and n must be greater than 1400.

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See also

• Almost prime
• Fermi–Dirac prime
• Perfect power
• Semiprime

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Further reading

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

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