Divisibility properties of the Fibonacci entry point

TL;DR

This paper proves a conjecture relating prime divisibility properties of Fibonacci numbers to algebraic group theory and Galois theory, providing a formula for the density of primes with specific divisibility conditions.

Contribution

It confirms Bruckman and Anderson's conjecture by linking Fibonacci divisibility to algebraic groups and applying Galois theory and Chebotarev density theorem to compute prime densities.

Findings

Proves the conjecture on the density of primes dividing Fibonacci entries.

Establishes a connection between Fibonacci divisibility and algebraic group order.

Provides a method to compute prime densities using Galois theory.

Abstract

For a prime $p$, let $Z(p)$ be the smallest positive integer $n$ so that $p$ divides $F_{n}$, the $n$th term in the Fibonacci sequence. Paul Bruckman and Peter Anderson conjectured a formula for $\zeta(m)$, the density of primes $p$ for which $m | Z(p)$ on the basis of numerical evidence. We prove Bruckman and Anderson's conjecture by studying the algebraic group $G : x^{2} - 5y^{2} = 1$ and relating $Z(p)$ to the order of $\alpha = (3/2,1/2) \in G(\F_{p})$. We are then able to use Galois theory and the Chebotarev density theorem to compute $\zeta(m)$.