What is the first 11 digit prime of Fibonacci?

Question

I am having difficulty finding the first eleven-digit prime number of Fibonacci..

If anyone has an answer I would greatly appreciate it. I'm mostly asking this because it's one part of a greater puzzle I am trying to solve..

Answer 1

Here are the Fibonacci primes:

Select[Table[Fibonacci[i], {i, 1, 200}], PrimeQ]

$\{2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437,$

$ 2971215073, 99194853094755497, 1066340417491710595814572169,$

$ 19134702400093278081449423917 \}$

Answer 2

It is a standard result that if the $n$-th Fibonacci number is prime, then $n$ is prime, unless $n=4$. This makes the search much easier.

The Fibonacci number $F_{47}$ has 10 digits, and the Fibonacci number $F_{59}$ has 12 digits. (this is easy to check by direct computation, or using the approximation $F_n \approx \phi^n / \sqrt{5}$.) There is only one prime between $47$ and $59$, namely $53$.

So all we have to check is whether $F_{53}$ is prime. But with a computer, it is easy to find that $953$ is a factor.

We can conclude that there are no 11-digit Fibonacci primes.