A Gaussian prime is a non-unit Gaussian integer divisible only by its associates and by the units (), and by no other Gaussian integers.
The Gaussian primes fall into one of three categories:
- Gaussian integers with imaginary part zero and a prime real part with a real prime satisfying (numbers of A002145 multiplied by or ).
- Gaussian integers with real part zero and an imaginary part with a real prime satisfying (numbers of A002145 multiplied by or ).
- Gaussian integers having both real and imaginary parts, and its complex norm is a real prime number in . (For example, given the numbers listed in A069003, the numbers are Gaussian primes.)
For example, neither 2 nor 5 are Gaussian primes, but since , which is a prime in , then is a Gaussian prime, divisible only by itself (or its associates , , ) and 1 (or its associates , and ) 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 given by .
- Nor is 29 is a Gaussian prime either since .