Funky definitions of prime number

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This article lists some definitions of prime number.

Definitions in terms of arithmetic functions

Divisor count function

A natural number is prime if and only if where is the divisor count function.

Divisor sum function

A natural number is prime if and only if , where is the divisor sum function.

Divisor power sum function

For any , a natural number is prime if and only if , where is the divisor power sum function.

Euler phi-function

A natural number is prime if and only if , where is the Euler phi-function.

Dedekind psi-function

A natural number is prime if and only if , where is the Dedekind psi-function.

von Mangoldt function

A natural number is prime if and only if and where is the von Mangoldt function.

In terms of facts true for prime numbers

Wilson's theorem

A natural number is prime if and only if and .

In algebraic terms

Groups

A natural number is prime if and only if it satisfies the following equivalent conditions:

  • The subgroup is a maximal subgroup in the group of integers.
  • The group of integers modulo is a simple group.

Rings and fields

A nautral number is prime if and only if it satisfies the following equivalent conditions:

  • The ideal is a maximal ideal in the ring of integers.
  • The ring of integers modulo is a field.

In terms of primality tests

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