Values of Lucas sequences modulo primes
type: Item
instance of: scholarly article
title: Values of Lucas sequences modulo primes
zbMATH Open document ID: 1076.11009
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-05-27T00:00:00Z
review text: Let \(F_n\) denote the Fibonacci numbers. It is very easy to show that primes \(p \neq 5\) divide \(F_{p \pm 1}\), but finding the correct sign involves a piece of quadratic reciprocity. \textit{M. Ward} [Pac. J. Math. 11, 379--386 (1961; Zbl 0112.26904)] has observed that the question which primes divide \(F_{(p \pm 1)/k}\) for \(k = 2, 3, 4 \ldots\) is connected with higher reciprocity laws [see also e.g. \textit{E. Lehmer}, Fibonacci Q. 4, 135--138 (1966; Zbl 0144.27405) or \textit{A. Aigner}, J. Reine Angew. Math. 274--275, 139--140 (1975; Zbl 0302.12002)]. In this article, these criteria are extended to general Lucas sequences, and in some cases the exact (nonzero) values of the terms modulo \(p\) are determined.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11A15
Mathematics Subject Classification ID: 11B50
zbMATH DE Number: 2067069
zbMATH Keywords: Fibonacci numbers
zbMATH Keywords: Lucas numbers
zbMATH Keywords: quartic reciprocity
author: Zhihong Sun
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069947
OpenAlex ID: W1967503415
cites work: Q4382830
cites work: Q3939864
cites work: On the theory of cubic residues and nonresidues
cites work: Supplements to the theory of quartic residues
cites work: Q4031910
cites work: Quartic residues and binary quadratic forms
cites work: Fibonacci numbers and Fermat's last theorem
DOI: 10.1216/RMJM/1181069947
description: scientific article; zbMATH DE number 2067069