Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers

Walter Carlip, and Lawrence Somer. "Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers." Colloquium Mathematicae 108.1 (2007): 73-93. <http://eudml.org/doc/284137>.

@article{WalterCarlip2007,
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-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. },
author = {Walter Carlip, Lawrence Somer},
journal = {Colloquium Mathematicae},
keywords = {Lucas -pseudoprime; Carmichael-Lucas numbers; algorithms},
language = {eng},
number = {1},
pages = {73-93},
title = {Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers},
url = {http://eudml.org/doc/284137},
volume = {108},
year = {2007},
}

TY - JOUR
AU - Walter Carlip
AU - Lawrence Somer
TI - Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers
JO - Colloquium Mathematicae
PY - 2007
VL - 108
IS - 1
SP - 73
EP - 93
AB - 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.
LA - eng
KW - Lucas -pseudoprime; Carmichael-Lucas numbers; algorithms
UR - http://eudml.org/doc/284137
ER -