Is there a composite number that satisfies these conditions? 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?
 A: This looks sort of interesting. I don't have an answer, but just a few observations. Instead of restricting to $3 + 2i$, we might consider the same condition holding for every $\alpha = a + bi$ prime to $N$, i.e., such that 
$$\alpha^N \equiv \bar{\alpha} \pmod N.$$
We are then led to consider a Gaussian analogue of Carmichael numbers, i.e., Carmichael ideals for the Gaussian integers, generated by numbers $N$ of the particular form $N = q_1 q_2\ldots q_k$ where $q_i \equiv 3\pmod 4$ for $i = 1, \ldots, k$. 
These will be ordinary Carmichael numbers $N$ but with the extra condition that 
$$(q_i^2 - 1)\; |\; N^2-1$$ 
for $i = 1, \ldots, k$. For example, the ordinary Carmichael number $7 \cdot 19 \cdot 67 = 8911$ fails to meet this stronger condition. 
I expect these stronger Carmichael numbers exist, but as I say I don't have an example. If I were researching this problem myself, I would try to get hold of a table of Carmichael numbers, see which ones have all of its prime factors $q_i$ congruent to 3 modulo 4, and then test the condition $q_i^2 - 1 \; |\; N^2-1$ on those.
A: 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). 
