Gaussian prime

A Gaussian primeMathworldPlanetmath p is a Gaussian integerMathworldPlanetmath a+bi (where i is the imaginary unitMathworldPlanetmath 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 partsMathworldPlanetmath 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 ( that since real primes of the form p=4n+1 can be represented as x2+y2, then in the complex plane they have the factorization (x+yi)(x-yi). For example, 17=42+12, 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.


  • 1 Kogbetliantz, Ervand George Handbook of first complex prime numbers London: Gordon and Breach Science Publishers (1971)
Title Gaussian prime
Canonical name GaussianPrime
Date of creation 2013-03-22 16:54:13
Last modified on 2013-03-22 16:54:13
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 7
Author PrimeFan (13766)
Entry type Definition
Classification msc 11R04
Classification msc 11A41