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.



Bound on the number of primitive divisors of the $n$th Fibonacci number
A known bound for the average $\frac{1}{x}\sum\_{n\leq x}P\_n$ is
$$
\limsup\_{x\rightarrow\infty} \frac{\log x}{x^2} \sum\_{n\leq x}P\_n
\leq
\frac{3}{2 \pi^2} \log \alpha.
$$
This is not much better than the above point-wise bound.
Making progress on this average seems to be hard. One could try to get some sort of prime number theorem
$$
\sum\_{n\leq x}P\_n \sim \,\,\,?
$$
The main obstacle here seems to be that all proofs of the prime number theorem (at least the ones I know) somehow rely on 
$$
\sum\_{p\leq x}\log p \sim \sum\_{n\leq x} \Lambda(n) 
$$
with $\Lambda$ denoting the von Mangoldt function. In our situation we can determine the asymptotic behavior of the following sums
$$
\sum\_{p: p|F\_i \text{ for some } i\leq x} e\_p\log p
 \sim 
\sum\_{n: n|F\_i \text{ for some } i\leq x} \Lambda(n) 
$$
where the sums are taken over (prime) divisors of Fibonacci numbers and $e\_p$ denotes the largest exponent such that whenever $p|F\_i\Rightarrow p^{e\_p}|F\_i$. The hard part is now, to relate this asymptotic behavior to that of
 $$
\sum\_{p: p|F\_i \text{ for some } i\leq x} \log p
$$
a weighted version of the sum we are interested in. It is not clear how this can be done.
For the averages of this weighted version it is known that  

$$
\limsup\_{x\rightarrow\infty} \frac{1}{x^2} \sum\_{p|F\_i\text{ for some } i\leq x} \log p 
\leq
\frac{3}{ \pi^2} \log \alpha.
$$
We also know that any improvement on the constant here would imply the existence of infinitely many Wall-Sun-Sun primes. It is an open problem whether such primes exist. However, since partial summation works differently here, the relation of the weighted average to the average is not obvious. 


