Funky definitions of prime number

This article lists some definitions of prime number.

Definitions in terms of arithmetic functions

Divisor count function

A natural number $n$ is prime if and only if ${\sigma }_0(n)=2$ where ${\sigma }_0$ is the divisor count function.

Divisor sum function

A natural number $n$ is prime if and only if $\sigma (n)=n+1$, where $\sigma$ is the divisor sum function.

Divisor power sum function

For any $k$, a natural number $n$ is prime if and only if $\sigma (n)=n^k+1$, where ${\sigma }_k$ is the $k^{th}$ divisor power sum function.

Euler phi-function

A natural number $n$ is prime if and only if $\phi (n)=n-1$, where $\phi$ is the Euler phi-function.

Dedekind psi-function

A natural number $n$ is prime if and only if $\psi (n)=n+1$, where $\psi$ is the Dedekind psi-function.

von Mangoldt function

A natural number $n$ is prime if and only if $n>1$ and $\mathrm{\Lambda }(n)=\log n$ where $\mathrm{\Lambda }$ is the von Mangoldt function.

In terms of facts true for prime numbers

Wilson's theorem

A natural number $n$ is prime if and only if $n>1$ and $(n-1)!\equiv -1(\mathrm{mod}n)$.

In algebraic terms

Groups

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

• The subgroup $n\mathbb{Z}$ is a maximal subgroup in the group of integers.
• The group of integers modulo $n$ is a simple group.

Rings and fields

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

• The ideal $n\mathbb{Z}$ is a maximal ideal in the ring of integers.
• The ring of integers modulo $n$ is a field.

In terms of primality tests

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