On equivalence classes of generalized Fibonacci sequences

Aoki, Miho ; Sakai, Yuho

Journal of integer sequences, Tome 19 (2016) no. 2

Résumé

We consider a generalized Fibonacci sequence $( G_{n} )$ by $G_1, G_2 \in \mathbb{Z} $ and $G_{n} = G_{n-1} + G_{n-2}$ for any integer $n$. Let $p$ be a prime number and let $d(p)$ be the smallest positive integer $n$ which satisfies $p \mid F_n$. In this article, we introduce equivalence relations for the set of generalized Fibonacci sequences. One of the equivalence relations is defined as follows. We write $( G_n ) \sim^* (G'_n )$ if there exist integers $m$ and $n$ satisfying $G_{m+1}G'_n \equiv \modd{G'_{n+1}G_m} {p}$. We prove the following: if $p \equiv 2$ (mod 5), then the number of equivalence classes $\overline{ ( G_n )}$ satisfying $p \nmid G_n$ for any integer $n$ is $(p+1)/d(p)-1$. If $p \equiv \pm 1$ (mod 5), then the number is $(p-1)/d(p)+1$. Our results are refinements of a theorem given by Kôzaki and Nakahara in 1999. They proved that there exists a generalized Fibonacci sequence $( G_{n} )$such that $p \nmid G_n$ for any $n \in \mathbb{Z} $ if and only if one of the following three conditions holds: (1) $p = 5$; (2) $p \equiv \pm 1$ (mod 5); (3) $p \equiv 2$ (mod 5) and $d(p)$.

Détail

Classification : 11B39
Keywords: Fibonacci number, Lucas number, generalized Fibonacci sequence