On the discriminator of Lucas sequences

TL;DR

This paper investigates the discriminator function of Lucas sequences, proving that for each fixed parameter, the discriminator exhibits a simple, predictable pattern beyond a certain sequence index, with a notable difference at k=1.

Contribution

The paper establishes a unified characterization of the discriminator function for Lucas sequences, highlighting a fundamental distinction in behavior when k=1 versus k>1.

Findings

Discriminator function stabilizes and becomes predictable after a certain index n_k.

For k=1, the discriminator behavior differs fundamentally from other k values.

The paper provides a proof of the simple characterization for all k≥1 beyond n_k.

Abstract

We consider the family of Lucas sequences uniquely determined by $U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),$ with initial values $U_0(k)=0$ and $U_1(k)=1$ and $k\ge 1$ an arbitrary integer. For any integer $n\ge 1$ the discriminator function $\mathcal{D}_k(n)$ of $U_n(k)$ is defined as the smallest integer $m$ such that $U_0(k),U_1(k),\ldots,U_{n-1}(k)$ are pairwise incongruent modulo $m$. Numerical work of Shallit on $\mathcal{D}_k(n)$ 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 $k\ge 1$ there is a constant $n_k$ such that ${\mathcal D}_{k}(n)$ has a simple characterization for every $n\ge n_k$. The case $k=1$ turns out to be fundamentally different from the case $k>1$.