Gaussian primes

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Distribution the Gaussian primes

A Gaussian prime is a non-unit Gaussian integer $m+ni$ divisible only by its associates and by the units ($1,i,-1,-i$), and by no other Gaussian integers.

The Gaussian primes fall into one of three categories:

1. Gaussian integers with imaginary part zero and a prime real part $m$ with $|m|$ a real prime satisfying $|m|\equiv 3mod4$ (numbers of A002145 multiplied by $1$ or $-1$).
2. Gaussian integers with real part zero and an imaginary part $n$ with $|n|$ a real prime satisfying $|n|\equiv 3mod4$ (numbers of A002145 multiplied by $i$ or $-i$).
3. Gaussian integers having both real and imaginary parts, and its complex norm is a real prime number in $\mathbb{\mathbb{Z}}$. (For example, given the numbers listed in A069003, the numbers $n\pm a(n)i$ are Gaussian primes.)

For example, neither 2 nor 5 are Gaussian primes, but since $2^2+5^2=29$, which is a prime in $\mathbb{\mathbb{Z}}$, then $2+5i$ is a Gaussian prime, divisible only by itself (or its associates $-2-5i$, $5-2i$, $-5+2i$) and 1 (or its associates $-1$, $i$ and $-i$) but no other Gaussian integers.

It should be noted that although all Gaussian primes in category 1 above are in A000040, 2 and all primes congruent to 1 mod 4 (Pythagorean primes) are Gaussian composites, since they are the sum of two squares, with factorization in $\mathbb{ℂ}$ given by $a^2+b^2=(a-bi)(a+bi)$.

See also

• Eisenstein primes