Vol. 9, No. 1, 2020

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Complete generalized Fibonacci sequences modulo primes

Mohammad Javaheri and Nikolai A. Krylov

Vol. 9 (2020), No. 1, 1–15
Abstract

We study generalized Fibonacci sequences Fn+1 = PFn QFn1 with initial values F0 = 0 and F1 = 1. Let P,Q be relatively prime nonzero integers such that P2 4Q is not a perfect square. We show that if Q = ±1 then the sequence {Fn}n=0 misses a congruence class modulo every large enough prime. If Q ± 1, we prove under the GRH that the sequence {Fn}n=0 hits every congruence class modulo infinitely many primes.

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