Let $F(n)$ be $n$-th Fibonacci number.$$F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, F(4) = 3 \text{ and so on. }$$Given a positive integer $n \gt 2$,Find the smallest prime number $P$ such that $P$ divides $F(n)$ but it does not divide any $F(k)$ smaller than $F(n)$ ?
Now,my question: Is it possible to find the answer without actually computing the value of $F(n)$?
I am looking for an interesting algorithm for this purpose.