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Abstract

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 $(N - \varepsilon(N))/d$, where the signature $\varepsilon(N) = (\frac{D}{N})$ 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 $(N - \varepsilon(N))/d$ 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 $4\nmid d$, 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.