On the discriminator of Lucas sequences

Abstract

We consider the family of Lucas sequences uniquely determined by Un+2(k) = (4k + 2)Un+1(k) − Un(k), with initial values U0(k) = 0 and U1(k) = 1 and k ≥ 1 an arbitrary integer. For any integer n ≥ 1 the discriminator function D k (n) of Un(k) is defined as the smallest integer m such that U0(k), U1(k), . . . , Un−1(k) are pairwise incongruent modulo m. Numerical work of Shallit on D k (n) suggests that it has a relatively simple characterization. In this paper we will prove that this is indeed the case by showing that for every k ≥ 1 there is a constant n k such that D k (n) has a simple characterization for every n ≥ n k . The case k = 1 turns out to be fundamentally different from the case k > 1.

Binary recurrences for which powers of two are discriminating moduli

“…We have u 2n = u n v n by the Binet formulas (4) and (5). We are now ready to show that z(2 k ) = 2 k .…”

“…By induction, we get that 2 v n for all n ≥ 0. The Binet formula for v n is (5) v n = α n + β n for all n ≥ 0.…”

Browkin’s discriminator conjecture

“…Since it is easy to show that 3 ∤ D q (n), it would be actually enough to study those integers d with 3 ∤ d (in which case the Browkin-Sȃlȃjan sequence is purely periodic modulo d). However, for completeness, we discuss the periodicity of the Browkin-Sȃlȃjan sequence for every d. In the sequel it is helpful to have in mind the trivial observation that, if 3 ∤ m, then (6) 2 ord 4m (9) = lcm(2, ord 4m (3)).…”

“…It provides, at the same time, the first instance of the determination of the discriminator for an infinite family of second-order recurrences having characteristic equation with rational roots. Very recently, Faye, Luca and Moree [6] determined the discriminator for another infinite family, this time having irreducible characteristic equation. Despite structural similarities, there are considerable differences in the details of the proofs in [6] and ours.…”

“…Very recently, Faye, Luca and Moree [6] determined the discriminator for another infinite family, this time having irreducible characteristic equation. Despite structural similarities, there are considerable differences in the details of the proofs in [6] and ours. For e.g., in our case it is much harder to exclude small prime numbers as discriminator values.…”