Code:

{
tst(n,a)=local(A=a^2+a);
kronecker(A,n)==-1&&
kronecker(a,n)==-1&&
Mod(A,n)^((n-1)/2)==-1&&
Mod(a,n)^((n-1)/2)==-1&&
Mod(Mod(z,n),z^2-(4*a/(a-1)-2)*z+1)^((n+1)/2)==-1
}

I am running this test against Richard Pinch's Carmichael number list and with David Broadhurst's CRT Semi-prime Pari/GP script.

The latter has only produced unique solutions for a given n, making 2 rounds with different a's sufficient for a primality proof?!?!?

Those pesky primes! I just found 2 solutions for 2728624939. However gcd(a

_{1}^2-a

_{2}^2,n)>1

I have now found n with 3 or 4 solutions too, and gcd(a

_{i}^2-a

_{j}^2,n)>1 for i != j.

I am seeing the same sort of GCDs for A=a+1 and kronecker(A,n)==-1. Hence the corresponding test can be done for suitable {a, a+1} and {a+1,a+2} which reduces the number of sub-tests, and no GCD need be calculated.