Vipul at 15:19, 6 May 2009

2009-05-06T15:19:47Z

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 A natural number <math>n</math> is prime if and only if <math>n > 1</math> and <math>(n - 1)! \equiv -1 \pmod n</math>.
 ==In terms of primality tests==
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Vipul: Created page with 'This article lists some definitions of prime number. ==Definitions in terms of arithmetic functions== ===Divisor count function=== A natural number $n$ is prime if ...'

2009-05-06T15:15:48Z

Created page with 'This article lists some definitions of prime number. ==Definitions in terms of arithmetic functions== ===Divisor count function=== A natural number <math>n</math> is prime if ...'

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This article lists some definitions of prime number.

==Definitions in terms of arithmetic functions==

===Divisor count function===

A natural number <math>n</math> is prime if and only if <math>\sigma_0(n) = 2</math> where <math>\sigma_0</math> is the [[divisor count function]].

===Divisor sum function===

A natural number <math>n</math> is prime if and only if <math>\sigma(n) = n + 1</math>, where <math>\sigma</math> is the [[divisor sum function]].

===Divisor power sum function===

For any <math>k</math>, a natural number <math>n</math> is prime if and only if <math>\sigma(n) = n^k + 1</math>, where <math>\sigma_k</math> is the <math>k^{th}</math> [[divisor power sum function]].

===Euler phi-function===

A natural number <math>n</math> is prime if and only if <math>\varphi(n) = n - 1</math>, where <math>\varphi</math> is the [[Euler phi-function]].

===Dedekind psi-function===

A natural number <math>n</math> is prime if and only if <math>\psi(n) = n + 1</math>, where <math>\psi</math> is the [[Dedekind psi-function]].

===von Mangoldt function===

A natural number <math>n</math> is prime if and only if <math>n > 1</math> and <math>\Lambda(n) = \log n</math> where <math>\Lambda</math> is the [[von Mangoldt function]].

==In terms of facts true for prime numbers==

===Wilson's theorem===

A natural number <math>n</math> is prime if and only if <math>n > 1</math> and <math>(n - 1)! \equiv -1 \pmod n</math>.

==In terms of primality tests==

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Funky definitions of prime number - Revision history

Vipul at 15:19, 6 May 2009

Vipul: Created page with 'This article lists some definitions of prime number. ==Definitions in terms of arithmetic functions== ===Divisor count function=== A natural number n is prime if ...'

Vipul: Created page with 'This article lists some definitions of prime number. ==Definitions in terms of arithmetic functions== ===Divisor count function=== A natural number $n$ is prime if ...'