It is a result of Carmichael that for any integer $n > 12$, the Fibonacci number $F_n$ has at least one primitive divisor, that is, a prime factor $p$ such that $p$ does not divide any $F_m$ with $1 \leq m < n$. Indeed, a more general result for all Lucas and Lehmer sequences has been proven by Bilu, Hanrot, and Voutier [1].

But my question is: **What about an upper bound for the number of primitive divisors of $F_n$?**

If $p$ is a primitive divisor of $F_n$, then it is known that $p \equiv \pm 1 \bmod n$, hence $p \geq n - 1$. Thus, if $P_n$ is the number of primitive divisors of $F_n$, we have

$$(n - 1)^{P_n} \leq \prod_{p \text{ prim. div. of } F_n} p \mid F_n \leq \alpha^{n-1} ,$$

where $\alpha := (1 + \sqrt{5})/2$, and consequently

$$P_n \leq \frac{(n - 1)\log \alpha}{\log (n - 1)} .$$

Is some better upper bound known?

[1] Bilu, Hanrot, and Voutier, *Existence of primitive divisors of Lucas and Lehmer numbers*, Journal für die reine und angewandte Mathematik (Crelles Journal), 2001(539), pp. 75-122.