Gaussian Integer

A Gaussian integer is a complex number where and are integers. The Gaussian integers are members of the imaginary quadratic field and form a ring often denoted , or sometimes (Hardy and Wright 1979, p. 179). The sum, difference, and product of two Gaussian integers are Gaussian integers, but divides iff there are integers and such that

(1)

(Shanks 1993).

Nonzero Gaussian integers have unique factorization into Gaussian primes, up to powers of and rearrangements.

The units of are and .

One definition of the norm of a Gaussian integer is its complex modulus

(2)

Another common definition (e.g., Herstein 1975; Hardy and Wright 1979, p. 182; Artin 1991, Dummit and Foote 2004) defines the norm of a Gaussian integer to be

(3)

the square of the above quantity. The latter definition is integer-valued, as required for the Euclidean function in the definition of a Euclidean ring. Because of the two possible definitions, caution is needed when consulting the literature.

For two independently and uniformly chosen Gaussian integers and of complex modulus at most , the limiting probability as that they are relatively prime is

(4)

(OEIS A088454), where is Catalan's constant (Collins and Johnson 1989, Pegg 2003; Finch 2003, p. 601).

Every Gaussian integer is within of a multiple of a nonzero Gaussian integer .

The plots above show roots of the Gaussian integers for various rational values of (Trott 2004, p. 24).