Gaussian Prime

Gaussian primes are Gaussian Integers $a+ib$ for which the norm $n(a+ib)=a^2+b^2$ is Prime or, if $b=0$, $a$ is a Prime Integer such that $\vert a\vert\equiv 3\ \left({{\rm mod\ } {4}}\right)$. The above plot of the Complex Plane shows the Gaussian primes as filled squares. The Gaussian primes with $\vert a\vert,\vert b\vert\leq 5$ are given by $-5-4i$, $-5-2i$, $-5+2i$, $-5+4i$, $-4-5i$, $-4-i$, $-4+i$, $-4+5i$, $-3-2i$, $-3$, $-3+2i$, $-2-5i$, $-2-3i$, $-2-i$, $-2+i$, $-2+3i$, $-2+5i$, $-1-4i$, $-1-2i$, $-1-i$, $-1+i$, $-1+2i$, $-1+4i$, $-3i$, $3i$, $1-4i$, $1-2i$, $1-i$, $1+i$, $1+2i$, $1+4i$, $2-5i$, $2-3i$, $2-i$, $2+i$, $2+3i$, $2+5i$, $3-2i$, $3$, $3+2i$, $4-5i$, $4-i$, $4+i$, $4+5i$, $5-4i$, $5-2i$, $5+2i$, $5+4i$.

See also Eisenstein Integer, Gaussian Integer

References

Guy, R. K. ``Gaussian Primes. Eisenstein-Jacobi Primes.'' §A16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 33-36, 1994.

Wagon, S. ``Gaussian Primes.'' §9.4 in Mathematica in Action. New York: W. H. Freeman, pp. 298-303, 1991.