$F_p,L_p$ both prime?

Question

It is well known that if the $n$ th Fibonacci number is a prime then it follows $n$ must also be a prime.

So we wonder if $F_p $ is prime or not. It is believed there are infinitely many Fibonacci primes. It is also believed there are infinitely many prime Lucas numbers ( or Lucas primes ).

So I wonder , are there many primes $p$ such that both $F_p $ and $L_p$ are prime ?

I have not checked mod 100.

Since fibonacci and Lucas numbers are related I wondered about that.

I know $ L_q = 1 \mod q $ for every odd prime $q$. Not sure if that is related.

Answer 1

In fact, the numbers are related : $$L_n=F_{n-1}+F_{n+1}$$ The list of known prime numbers of both kinds reveals that we have a prime pair $(F_n,L_n)$ for the positive integers $$4\ 5\ 7\ 11\ 13\ 17\ 47$$ as well as the surprising pair for $n=148\ 091$ for which however $F_n$ is "only" a probable prime.

For both kinds of numbers, we can expect infinite many primes because of the relative slow growth rate (only exponential), but I would not expect another pair. I am already very surprised of the huge pair above (more than $30\ 000$ digits each !)