TITLE: Is there a composite number that satisfies these conditions?
QUESTION [13 upvotes]: We know that if $q=4k+3$ ($q$ is a prime), then $(a+bi)^q=a-bi \pmod q$ for every Gaussian integer $a+bi$. Now consider a composite number $N=4k+3$ that satisfies this condition for the case $a+bi=3+2i$. I use Mathematica 8 and find no solution less than $5\cdot 10^7$. Can someone find a larger number for the condition, and can this be used for a primality test?

REPLY [5 votes]: As Mr. R. Gerbicz pointed out in the mersenne forum an eventual counterexample for the base 3+2i must be 13-PRP (just multiply the equation with its conjugate). The first point to check is to make a list of pseudoprimes base 13 which are 3 (mod 4). I checked them to 10^10 and there is no counterexample which pass this test (a couple of them which are 1 (mod 4) passes the complex base test, but none of the 3 (mod 4)). However, the general opinion is that this test is a "hidden" multi-base PRP test, or a (n-1)(n+1) combined test, and as Mr. Tom Womack pointed out in that thread, if a couterexample exists, it must be HIGH (somewhere in 10^30 or so).