we know that if $q=4k+3$ ($q$ is a prime), then $(a+bI)^q=a^q+b^q(I)^{4k+3} =a -bI(mod q)$ for every gaussian integer $(a+bi)$ ,Now consider a composite $N=4k+3$ satisfies this condistion for $a+bi=3+2i$, I use Mathematica8 and find no solutison$ less than $5\cdots 10^7$, can someone find a lager number for the condition . I guess it's impossible for a composite N.So this can be use for Deterministic Primality test .