The GCD Sequences of the Altered Lucas Sequences

On GCD-morphic sequences

This note is a response to one of the problems posed by Kwa´sniewski in [1, 2], see also [3] i.e. GCD-morphic Problem III. We show that any GCD-morphic sequence F is at the point product of primary GCD-morphic sequences and any GCD-morphic sequence is encoded by natural number valued sequence satisfying condition (C1). The problem of general importance-for example in number theory was formulate...

On Lucas Sequences Computation

This paper introduces an improvement to a currently published algorithm to compute both Lucas “sister” sequences Vk and Uk. The proposed algorithm uses Lucas sequence properties to improve the running time by about 20% over the algorithm published in [1].

On generalized Lucas sequences

We introduce the notions of unsigned and signed generalized Lucas sequences and prove certain polynomial recurrence relations on their characteristic polynomials. We also characterize when these characteristic polynomials are irreducible polynomials over a finite field. Moreover, we obtain the explicit expressions of the remainders of Dickson polynomials of the first kind divided by the charact...

Congruences concerning Lucas Sequences

On the prime density of Lucas sequences

The density of primes dividing at least one term of the Lucas sequence defined by L0(P) = 2, L1(P)= P and Ln(P) = PLn-1(P) + Ln-2(P) for n ~ 2, with P an arbitrary integer, is determined.