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?

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?

We know that if $q=4k+3$ ($q$ is a prime) then $(a+bI)^q=a-bI$ (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 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?

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We know , that if $q=4k+3$ (q $q$ is a prime) ,$(a+bI)^q=a-bI$ then $(a+bI)^q=a-bI$ ($a+bI$ is a gaussian integer),now integer). Now consider a composite number $N=4k+3$ satisied that satisfies the condition. suposse Suposse $a+bI=3+2I$ ,a+bI=3+2I$. I use Mathematica8 and find no sloution exist solution less than 5*10^7.some one can 5*10^7. Can someone find large a larger number for the conditions and can this be used for primality test?

we know , if $q=4k+3$ (q is a prime) ,$(a+bI)^q=a-bI$ ($a+bI$ is a gaussian integer),now consider a composite number $N=4k+3$ satisied the condition. suposse $a+bI=3+2I$ ,I use Mathematica8 and find no sloution exist less than 5*10^7.some one can find large number for the conditions and can this used for primality test?