Funky definitions of prime number
This article lists some definitions of prime number.
Contents
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
