# Definition:Gaussian Prime

Jump to navigation
Jump to search

## Definition

### Definition 1

Let $x \in \Z \sqbrk i$ be a Gaussian integer.

$x$ is a **Gaussian prime** if and only if:

- it cannot be expressed as the product of two Gaussian integers, neither of which is a unit of $\Z \sqbrk i$ (that is, $\pm 1$ or $\pm i$)
- it is not itself a unit of $\Z \sqbrk i$.

### Definition 2

A **Gaussian prime** is a Gaussian integer which has exactly $8$ divisors which are themselves Gaussian integers.

## Examples

The following are examples of **Gaussian primes**:

- $1 + i$
- $4 - i$
- $7 + 2 i$

## Also see

- Results about
**Gaussian primes**can be found**here**.

## Source of Name

This entry was named for Carl Friedrich Gauss.

## Historical Note

The concept of a Gaussian prime was introduced by Carl Friedrich Gauss in his papers of $1828$ and $1832$.