Complete generalized Fibonacci sequences modulo primes

We study generalized Fibonacci sequences $F_{n + 1} = P F_{n} - Q F_{n - 1}$ with initial values $F_{0} = 0$ and $F_{1} = 1$ . Let $P, Q$ be relatively prime nonzero integers such that $P^{2} - 4 Q$ is not a perfect square. We show that if $Q = \pm 1$ then the sequence ${F_{n}}_{n = 0}^{\infty}$ misses a congruence class modulo every large enough prime. If $Q \neq \pm 1$ , we prove under the GRH that the sequence ${F_{n}}_{n = 0}^{\infty}$ hits every congruence class modulo infinitely many primes.

PDF Access Denied

We have not been able to recognize your IP address 13.59.236.219 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords

generalized Fibonacci sequence, complete sequence

Mathematical Subject Classification 2010

Primary: 11B39

Secondary: 11B50

Milestones

Received: 17 December 2018

Revised: 7 November 2019

Accepted: 22 November 2019

Published: 8 January 2020

Authors

Mohammad Javaheri
Department of Mathematics Siena College Loudonville, NY United States

Nikolai A. Krylov
Department of Mathematics Siena College Loudonville, NY United States

Vol. 9, No. 1, 2020

Mohammad Javaheri and Nikolai A. Krylov

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors