Relationship between Vieta-Lucas polynomials and Lucas sequences

TL;DR

This paper explores the relationship between Vieta-Lucas polynomials and Lucas sequences, establishing conditions for polynomial congruences based on divisibility properties of Lucas sequences.

Contribution

It provides a novel connection between Vieta-Lucas polynomials and Lucas sequences, offering new criteria for solving polynomial congruences modulo primes.

Findings

Solution existence characterized by Lucas sequence divisibility

Established congruence conditions involving gcd and sequence parameters

Linked polynomial roots to Lucas sequence divisibility properties

Abstract

Let $w_n=w_n(P,Q)$ be numerical sequences which satisfy the recursion relation \begin{equation*} w_{n+2}=Pw_{n+1}-Qw_n. \end{equation*} We consider two special cases $(w_0,w_1)=(0,1)$ and $(w_0,w_1)=(2,P)$ and we denote them by $U_n$ and $V_n$ respectively. Vieta-Lucas polynomial $V_n(X,1)$ is the polynomial of degree $n$. We show that the congruence equation $V_n(X,1)\equiv C \mod p$ has a solution if and only if $U_{(p-\epsilon)/d}(C+2,C+2)$ is divisible by $p$, where $\epsilon\in\{\pm 1\}$ depends on $C$ and $p$, and $d=\gcd(n,p-\epsilon)$.