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Prime divisors of the Lagarias sequence

Pieter Moree, Peter Stevenhagen (2001)

Journal de théorie des nombres de Bordeaux

We solve a 1985 challenge problem posed by Lagarias [5] by determining, under GRH, the density of the set of prime numbers that occur as divisor of some term of the sequence x n n = 1 ∞ defined by the linear recurrence x n + 1 = x n + x n - 1 and the initial values x 0 = 3 and x 1 = 1 . This is the first example of a ænon-torsionÆ second order recurrent sequence with irreducible recurrence relation for which we can determine the associated density of prime divisors.

Primefree shifted Lucas sequences

Lenny Jones (2015)

Acta Arithmetica

We say a sequence = ( s ₙ ) n ≥ 0 is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of . In this article, we focus on the particular Lucas sequences of the first kind, a = ( u ₙ ) n ≥ 0 , defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-2 for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences a ± k are simultaneously primefree. This result extends...

Primitive divisors of Lucas and Lehmer sequences, II

Paul M. Voutier (1996)

Journal de théorie des nombres de Bordeaux

Let α and β are conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all α and β with h ( β / α ) ≤ 4 , the n -th element of these sequences has a primitive divisor for n > 30 . In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.

Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers

Walter Carlip, Lawrence Somer (2007)

Colloquium Mathematicae

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...

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