# Funky 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

Fill this in later