A Gaussian prime is a Gaussian integer (where is the imaginary unit and and are real integers) that is divisible only by the units 1, , and , itself, its associates and no others. For example, is a Gaussian prime because there is no pair of Gaussian integers (besides the units and associates) that multiply to . But is not a Gaussian prime because . If is prime then so are , and , as well as the associates , , and .
The real and the imaginary parts must be of different parity. For a real prime to be a Gaussian prime of the form , the real part has to be of the form ; the same goes for the associates . It follows from Fermat’s theorem on sums of two squares (http://planetmath.org/RepresentingPrimesAsX2ny2) that since real primes of the form can be represented as , then in the complex plane they have the factorization . For example, , so .
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)
|Date of creation||2013-03-22 16:54:13|
|Last modified on||2013-03-22 16:54:13|
|Last modified by||PrimeFan (13766)|