Journal of Integer Sequences, Vol. 19 (2016), Article 16.2.6

On Equivalence Classes of Generalized Fibonacci Sequences

Miho Aoki and Yuho Sakai
Department of Mathematics
Shimane University
Matsue, Shimane 690-8504


We consider a generalized Fibonacci sequence ( Gn ) by $G_1, G_2 \in \mathbb{Z} $ and Gn = Gn-1 + Gn-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 ≡ 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 ≡ ± 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 ( Gn )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 ≡ ± 1 (mod 5); (3) p ≡ 2 (mod 5) and d(p)<p+1.

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Received November 7 2015; revised versions received January 18 2016; January 20 2016; January 25 2016. Published in Journal of Integer Sequences, February 5 2016.

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