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.