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arXiv · 1603.07863

On the Lucas Property of Linear Recurrent Sequences

Hao Zhong·Tianxin Cai
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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.

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Hao Zhong, Tianxin Cai. 2016-03-25. On the Lucas Property of Linear Recurrent Sequences. https://arxiv.org/abs/1603.07863

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