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Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers

100%

Carlip W., Somer L.

Colloquium Mathematicum

2007

tom 108

nr 1

73-93

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.

Unbounded stability of two-term recurrence sequences modulo $2^k$

100%

Carlip W., Jacobson E.

Acta Arithmetica

1996

tom 74

nr 4

329-346

Ograniczanie wyników

1 Acta Arithmetica

1 Colloquium Mathematicum

2 Carlip W.

1 Jacobson E.

1 Somer L.

1 2007

1 1996

Wyniki wyszukiwania

Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers

Unbounded stability of two-term recurrence sequences modulo $2^k$