As the Gaussian integers form a principal ideal domain they form also a unique factorization domain. This implies that a Gaussian integer is irreducible (that is, it is not the product of two non-units) if and only if it is prime (that is, it generates a prime ideal).

The prime elements of Z[i] are also known as Gaussian primes. An associate of a Gaussian prime is also a Gaussian prime. The conjugate of a Gaussian prime is also a Gaussian prime (this implies that Gaussian primes are symmetric about the real and imaginary axes).

A positive integer is a Gaussian prime if and only if it is a prime number that is congruent to 3 modulo 4 (that is, it may be written 4n + 3, with n a nonnegative integer) (sequence A002145 in the OEIS). The other prime numbers are not Gaussian primes, but each is the product of two conjugate Gaussian primes.

A Gaussian integer a + bi is a Gaussian prime if and only if either:

  • one of a, b is zero and absolute value of the other is a prime number of the form 4n + 3 (with n a nonnegative integer), or
  • both are nonzero and a2 + b2 is a prime number (which will not be of the form 4n + 3).

In other words, a Gaussian integer is a Gaussian prime if and only if either its norm is a prime number, or it is the product of a unit (±1, ±i) and a prime number of the form 4n + 3.

It follows that there are three cases for the factorization of a prime number p in the Gaussian integers:

  • If p is congruent to 3 modulo 4, then it is a Gaussian prime; in the language of algebraic number theory, p is said to be inert in the Gaussian integers.
  • If p is congruent to 1 modulo 4, then it is the product of a Gaussian prime by its conjugate, both of which are non-associated Gaussian primes (neither is the product of the other by a unit); p is said to be a decomposed prime in the Gaussian integers. For example, 5 = (2 + i)(2 − i) and 13 = (3 + 2i)(3 − 2i).
  • If p = 2, we have 2 = (1 + i)(1 − i) = i(1 − i)2; that is, 2 is the product of the square of a Gaussian prime by a unit; it is the unique ramified prime in the Gaussian integers.

 

size = 120;
table = Table[
If[PrimeQ[x + I*y], 0, 1], {x, -size, size}, {y, -size, size}];
Show[Graphics[Raster[table]], ColorFunction -> "TemperatureMap"]

Content retrieved from: https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_primes