I found 11 prime fibonacci numbers in about 11 seconds. It seemed to stick on the 12th one. I went to bed, thinking that I would have all 26 in the morning. When I woke up, it still hadn't found the 12th one. I was amazed at first, but then I started looking for a pre made list of prime fibonacci numbers on the internet. I found them at http://www.utm.edu/research/primes/lists/top_ten/topten.pdf page 88. Notice that only 26 of them have been found (at least since 2000).
Also notice the number of digits the last one has 2023. In order to do math on integers of those size you have to use a big number library. I started using NTL. It is an easy package to set up and use.
I want to write a program that will use some of the faster, albeit more complicated, primality tests. Then I will start searching for the next prime fibonacci number. I also need to write the program, so that it can be interrupted, and will continue where it left off. If I find the next fibonacci prime I may win some recognition with my geek friends!
One interesting tidbit I found while researching all of this is that, If F is the fibonacci function, then F(n) is prime only if n is prime! However, if n is prime F(n) is not always prime.