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2017 Mod $p$ equivalence classes of linear recurrence sequences of degree~$2$
Miho Aoki, Yuho Sakai
Rocky Mountain J. Math. 47(8): 2513-2533 (2017). DOI: 10.1216/RMJ-2017-47-8-2513

Abstract

Laxton introduced a group structure on the set of equivalence classes of linear recurrence sequences of degree~2. This result yields much information on the divisibilities of such sequences. In this paper, we introduce other equivalence relations for the set of linear recurrence sequences $(G_n)$, which are defined by $G_0, G_1 \in \mathbb{Z} $ and $G_n=TG_{n-1}-NG_{n-2}$ for fixed integers~$T$ and $N=\pm 1$. The relations are given by certain congruences modulo~$p$ for a fixed prime number~$p$, which are different from Laxton's without modulo $p$ equivalence relations. We determine the initial terms $G_0$ and $G_1$ of all of the representatives of the equivalence classes $\overline {(G_n)}$ satisfying $p\nmid G_n$ for any integer~$n$ and give the number of equivalence classes. Furthermore, we determine the representatives of Laxton's without modulo~$p$ classes from our modulo~$p$ classes.

Citation

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Miho Aoki. Yuho Sakai. "Mod $p$ equivalence classes of linear recurrence sequences of degree~$2$." Rocky Mountain J. Math. 47 (8) 2513 - 2533, 2017. https://doi.org/10.1216/RMJ-2017-47-8-2513

Information

Published: 2017
First available in Project Euclid: 3 February 2018

zbMATH: 06840986
MathSciNet: MR3760304
Digital Object Identifier: 10.1216/RMJ-2017-47-8-2513

Subjects:
Primary: 11B37, 11B39

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.47 • No. 8 • 2017
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