Let d be a fixed positive integer. A Lucas d-pseudoprime is a Lucas pseudoprime N for which there exists a Lucas sequence U(P,Q) such that the rank of appearance of N in U(P,Q) is exactly (N-ε(N))/d, where the signature ε(N) = (D/N) is given by the Jacobi symbol with respect to the discriminant D of U. A Lucas d-pseudoprime N is a primitive Lucas d-pseudoprime if (N-ε(N))/d is the maximal rank of N among Lucas sequences U(P,Q) that exhibit N as a Lucas pseudoprime.
We derive new criteria to bound the number of d-pseudoprimes. In a previous paper, it was shown that if 4 ∤ d, then there exist only finitely many Lucas d-pseudoprimes. Using our new criteria, we show here that if d = 4m, then there exist only finitely many primitive Lucas d-pseudoprimes when m is odd and not a square.
We also present two algorithms that produce almost every primitive Lucas d-pseudoprime with three distinct prime divisors when 4 | d and show that every number produced by these two algorithms is a Carmichael-Lucas number. We offer numerical evidence to support conjectures that there exist infinitely many Lucas d-pseudoprimes of this type when d is a square and infinitely many Carmichael-Lucas numbers with exactly three distinct prime divisors.