On the Lucas Property of Linear Recurrent Sequences

Abstract

We say that an arithmetical function $S:\mathbb{N}\to \mathbb{Z}$ 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 $0\le n_i\le p-1,n,n_i\in \mathbb{N}$. In this note, we discuss the Lucas property of Fibonacci sequences and Lucas numbers. Meanwhile, we find some other interesting results.

8 pages