Gaussian prime

A Gaussian prime $p$ is a Gaussian integer $a+bi$ (where $i$ is the imaginary unit and $a$ and $b$ are real integers) that is divisible only by the units 1, $-1$, $i$ and $-i$, itself, its associates and no others. For example, $3+20i$ is a Gaussian prime because there is no pair of Gaussian integers (besides the units and associates) that multiply to $3+20i$. But $3+21i$ is not a Gaussian prime because $3(-i)(1+i){(1+2i)}^2=3+21i$. If $a+bi$ is prime then so are $a-bi$, $-a+bi$ and $-a-bi$, as well as the associates $b+ai$, $b-ai$, $b-ai$ and $-b-ai$.

The real and the imaginary parts must be of different parity. For a real prime to be a Gaussian prime of the form $p+0i$, the real part has to be of the form $p=4n-1$; the same goes for the associates $0+pi$. It follows from Fermat’s theorem on sums of two squares (http://planetmath.org/RepresentingPrimesAsX2ny2) that since real primes of the form $p=4n+1$ can be represented as $x^2+y^2$, then in the complex plane they have the factorization $(x+yi)(x-yi)$. For example, $17=4^2+1^2$, so $(4+i)(4-i)=17$.

Sometimes Gaussian primes are simply called “complex primes,” which is an incorrect term found in some of the older literature.