On the Lucas Property of Linear Recurrent Sequences

TL;DR

This paper investigates the Lucas property in linear recurrent sequences, especially Fibonacci and Lucas numbers, providing new insights and results about their modular properties related to prime numbers.

Contribution

It characterizes the Lucas property for Fibonacci sequences and Lucas numbers, revealing new results and extending understanding of their modular behavior.

Findings

Fibonacci sequences exhibit the Lucas property under certain conditions.

Lucas numbers also display the Lucas property in specific cases.

New results connect the Lucas property with prime modular arithmetic.

Abstract

We say that an arithmetical function $S:\mathbb{N}\rightarrow\mathbb{Z}$ has Lucas property if for any prime $p$, \begin{equation*} S(n)\equiv S(n_{0})S(n_{1})\ldots S(n_{r})\pmod p, \end{equation*} where $n=\sum_{i=0}^{r}n_{i}p^{i}$, with $0 \leq n_{i} \leq p-1,n,n_{i}\in\mathbb{N}$. In this note, we discuss the Lucas property of Fibonacci sequences and Lucas numbers. Meanwhile, we find some other interesting results.