Exceptional real Lucas sequences

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...

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].

Congruences concerning Lucas Sequences

On squares in Lucas sequences

Let P and Q be non-zero integers. The Lucas sequence {Un(P,Q)} is defined by U0 = 0, U1 = 1, Un = PUn−1 − QUn−2 (n ≥ 2). The question of when Un(P,Q) can be a perfect square has generated interest in the literature. We show that for n = 2, ..., 7, Un is a square for infinitely many pairs (P,Q) with gcd(P,Q) = 1; further, for n = 8, ..., 12, the only non-degenerate sequences where gcd(P,Q) = 1 a...

Sum Relations for Lucas Sequences

V0 = 2, V1 = p, and Vn = PVn−1 −QVn−2 (n ≥ 2). (2) The characteristic equation x − Px + Q = 0 of the sequences Un and Vn has two roots α = (P + √ D)/2 and β = (P − √ D)/2 with the discriminant D = P 2 − 4Q. Note that D = α−β. Furthermore,D = 0 means x2−Px+Q = 0 has the repeated root α = β = P/2. It is well known that for any n ∈ N (see [4, pp. 41–44]), PUn + Vn = 2Un+1, (α− β)Un = α − β, Vn = α...