## Primitive Lucas $d$-pseudoprimes and Carmichael–Lucas numbers

### Volume 108 / 2007

#### 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.