Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers

• Autorzy: Walter Carlip , Lawrence Somer
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11A51: Factorization; primality
• 11B37: Recurrences
• 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
• 11A41: Primes
• 11Y11: Primality

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2007
• Tom: 108
• Numer: 1
• Strony: 73-93

Daty

wydano

2007

Twórcy

autor

Walter Carlip

• Department of Mathematics, Franklin and Marshall College, Lancaster, PA 17604, U.S.A.

autor

Lawrence Somer

• Department of Mathematics, Catholic University of America, Washington, DC 20064, U.S.A.

Identyfikatory

DOI

10.4064/cm108-1-7