Divisibility properties of the Fibonacci entry point

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 $m\mathrm{\mid }Z(p)$ 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 $\alpha =(3\mathrm{/}2,1\mathrm{/}2)\in G({\text{F}}_p)$. We are then able to use Galois theory and the Chebotarev density theorem to compute $\zeta (m)$.