Gaussian Integer

A Complex Number $a+bi$ where $a$ and $b$ are Integers. The Gaussian integers are members of the Quadratic Field ${\Bbb{Q}}(\sqrt{-1}\,)$. The sum, difference, and product of two Gaussian integers are Gaussian integers, but $a+bi\vert c+di$ only if there is an $e+fi$ such that

$$(a+bi)(e+fi)=(ae-bf)+(af+be)i=c+di.$$

Gaussian Integers can be uniquely factored in terms of other Gaussian Integers up to Powers of $i$ and rearrangements.

The norm of a Gaussian integer is defined by

$$n(x+iy)=x^2+y^2.$$

Gaussian Primes are Gaussian integers $a+ib$ for which $n(a+ib)=a^2+b^2$ is Prime and $a$ is a Prime Integer satisfying $a\equiv 3\ \left({{\rm mod\ } {4}}\right)$.

1. If $2\vert n(x+iy)$, then $1+i$ and $1-i \vert x+iy$. These factors are equivalent since $-i(i-1)=i+1$. For example, $2=(1+i)(1-i)$ is not a Gaussian Prime.

2. If $n(a+ib)\equiv 3\ \left({{\rm mod\ } {4}}\right)\vert n(x+iy)$, then $n(a+ib)\vert x+iy$.

3. If $n(a+ib)\equiv 1\ \left({{\rm mod\ } {4}}\right)\vert n(x+iy)$, then $a+ib$ or $b+ia\vert x+iy$. If both do, then $n(a+ib)\vert x+iy$.

Every Gaussian integer is within $\vert n\vert/\sqrt{2}$ of a multiple of a Gaussian integer $n$.

See also Complex Number, Eisenstein Integer, Gaussian Prime, Integer, Octonion

References

Conway, J. H. and Guy, R. K. ``Gauss's Whole Numbers.'' In The Book of Numbers. New York: Springer-Verlag, pp. 217-223, 1996.

Shanks, D. ``Gaussian Integers and Two Applications.'' §50 in Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 149-151, 1993.