1. Is it true that if a Fibonacci number $F\_{n}$ divides the product of two Fibonacci numbers, then it must divide at least one of them?
2. Is it true that for all $n \ne 1,2,6,12$, there exists a prime divisor $p$ of $F\_{n}$ such that the entry point (first appearance as a divisor in the Fibonacci sequence) of $p$ is at position $n$?


I can see that the second statement implies the first (at least for $n \ne 1,2,6,12$).


Also, Carmichael's theorem says that for all $n \ne 1,2,6,12$, there exists a prime divisor $p$ of $F\_{n}$ such that $p \nmid F\_k$ for (positive integer) $k \lt n$, and the second statement is equivalent to a sort of counterpart going in the other direction: 
for all $n \ne 1,2,6,12$, there exists a prime divisor $p$ of $F\_{n}$ such that $p \nmid F\_k$ for $k \gt n$ save only for the obvious exceptions where $k$ is a multiple of $n$ (since $F\_{n}\mid F\_{k}$ when $n \mid k$).


These seem like natural questions to ask but I haven't seen them addressed.



Is every Fibonacci number "Fibonacci-prime"?
As François Brunault said in his comment, your #2 is true: Carmichael's theorem, which establishes the existence of a *primitive prime divisor* of $F\_n$ for every $n\ne1,2,6,12$, together with [$\gcd(F\_m,F\_n) = F\_{\gcd(m,n)}$](http://en.wikipedia.org/wiki/Fibonacci_number#Primes_and_divisibility), shows that no Fibonacci numbers are divisible by a primitive prime divisor of $F\_n$ except $F\_{kn}$.


This gives an answer to #1 as well: suppose that $F\_n \mid F\_a F\_b$, and let $p$ be a primitive prime divisor of $F\_n$. Then $p \mid F\_a$ or $p\mid F\_b$, and so one of $a$ or $b$ must be a multiple of $n$, so that $F\_n\mid F\_a$ or $F\_n\mid F\_b$. This proof doesn't work for $n=1,2,6,12$, but $n=1,2$ are trivial, and the cases $n=6,12$ follow by considering the (periodic) Fibonacci sequence modulo $8$ and $144$, respectively. (In the latter case it may be easier to examine the periodic sequence $\gcd(F\_n,144) = F\_{\gcd(n,12)}$.)


Note that this proof gives the generalization that if $F\_n$ divides the product of any finite number of Fibonacci numbers, then it must divide one of them - except for $n=6$ and $n=12$ for which this is false!


