With respect to the enhanced Lucas test for n with parameters P and Q, using congruences for both of the Lucas sequences U_n and V_n,
I’ve come to the realization that the Lucas congruences are equivalent to a specific test inspired by the Frobenius endomorphism in the ring Z/nZ[sqrt(D)] where D=P^2-4Q.
Specifically, and assuming P and Q are chosen such that Jacobi(D,n)=-1,
U_{n+1} == 0 (mod n) and V_{n+1} == 2Q (mod n)
is equivalent to
x^n = conjugate(x) in Z/nZ[sqrt(D)] where x = (P+sqrt(D))/2 .
Equivalence of Enhanced Lucas Test with a Frobenius Test
