Bruckman–Anderson conjecture on the density of Fibonacci entry-point divisibility

Let $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 2$ be the Fibonacci sequence. For a prime $p$, let $Z(p)$ be the smallest positive integer $m$ such that $p\mid F_m$. For a positive integer $m$, define

$$M(m,x)=\#\{p\leq x:p\text{ is prime and }m\mid Z(p)\},$$

and

$$\zeta(m)=\lim_{x\to\infty}\frac{M(m,x)}{\pi(x)},$$

when this limit exists. For a prime power $q^e$ with $e\geq 1$, and for an arbitrary positive integer $m$, define

$$\rho(m)=\begin{cases}1&\text{if }10\nmid m,\\[2pt]\frac54&\text{if }m\equiv10\pmod{20},\\[2pt]\frac12&\text{if }20\mid m.\end{cases}$$

Bruckman–Anderson conjecture. If $m=q^e$ is a prime power with $e\geq 1$, then

$$\zeta(q^e)=\frac{q^{2-e}}{q^2-1}.$$

For an arbitrary positive integer $m$,

$$\zeta(m)=\rho(m)\prod_{q^j\parallel m}\zeta(q^j),$$

where the product is over all prime powers occurring in the prime factorization of $m$. The conjecture gives the expected density of primes whose Fibonacci entry point is divisible by $m$, refining the known density $2/3$ for even Fibonacci entry points. The source presents this as a conjecture based on numerical data; its resolution is not stated here.