# Complete Generalized Fibonacci Sequences Modulo Primes

## TL;DR

This paper investigates the distribution of generalized Fibonacci sequences modulo primes, showing that their residue classes are either missed or hit infinitely often depending on parameters, under certain number-theoretic assumptions.

## Contribution

It establishes conditions under which generalized Fibonacci sequences modulo primes either miss or hit all residue classes, extending understanding of their modular distribution.

## Key findings

- Sequences with Q=±1 miss some residue classes modulo large primes.
- Sequences with Q≠±1 hit all residue classes modulo infinitely many primes (assuming GRH).
- Results depend on the parameters P, Q and number-theoretic conjectures.

## 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.