We know that if $q=4k+3$ ($q$ is a prime) prime), then $(a+bI)^q=a-bI (a+bi)^q=a-bi \pmod q$ ($a+bI$ is a gaussian integer). for every Gaussian integer $a+bi$. Now consider a composite number $N=4k+3$ that satisfies the this condition . Suposse for the case $a+bI=3+2I$. a+bi=3+2i$. I use Mathematica 8 and find no solution less than 5*10^7$5\cdot 10^7$. Can someone find a larger number for the conditions condition, and can this be used for a primality test? 4 added 1 characters in body We know that if$q=4k+3$($q$is a prime) then$(a+bI)^q=a-bI \pmod q$($a+bI$is a gaussian integer). Now consider a composite number$N=4k+3$that satisfies the condition. Suposse$a+bI=3+2I\$. I use Mathematica8 Mathematica 8 and find no solution less than 5*10^7. Can someone find a larger number for the conditions and can this be used for primality test?