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

Walter Carlip¹ ; Lawrence Somer²

¹ Department of Mathematics Franklin and Marshall College Lancaster, PA 17604, U.S.A.
² Department of Mathematics Catholic University of America Washington, DC 20064, U.S.A.

Colloquium Mathematicum, Tome 108 (2007) no. 1, pp. 73-93

Résumé

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.

Détail

DOI : 10.4064/cm108-1-7

Keywords: fixed positive integer lucas d pseudoprime lucas pseudoprime which there exists lucas sequence rank appearance exactly varepsilon where signature varepsilon frac given jacobi symbol respect discriminant lucas d pseudoprime primitive lucas d pseudoprime varepsilon maximal rank among lucas sequences exhibit lucas pseudoprime derive criteria bound number d pseudoprimes previous paper shown nmid there exist only finitely many lucas d pseudoprimes using criteria here there exist only finitely many primitive lucas d pseudoprimes odd square present algorithms produce almost every primitive lucas d pseudoprime three distinct prime divisors every number produced these algorithms carmichael lucas number offer numerical evidence support conjectures there exist infinitely many lucas d pseudoprimes type square infinitely many carmichael lucas numbers exactly three distinct prime divisors