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arXiv · 1812.01048

Complete Generalized Fibonacci Sequences Modulo Primes

Mohammad Javaheri·Nikolai Krylov
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Abstract

We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ 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 prime large enough. On the other hand, if $Q \neq \pm 1$, we prove that (under GRH) the sequence $\{F_n\}_{n=0}^\infty$ hits every congruence class modulo infinitely many primes.

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Mohammad Javaheri, Nikolai Krylov. 2018-12-03. Complete Generalized Fibonacci Sequences Modulo Primes. https://doi.org/10.2140/moscow.2020.9.1

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