Definition:Gaussian Prime

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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 $1832$.