Fibonacci Number with Prime Index is not necessarily Prime

Theorem
Let $p \in \Z_{>0}$ be a prime number.

Let $F_p$ be the $p$th Fibonacci number.

Then $F_p$ is not itself necessarily prime.

Proof
Proof by Counterexample:

$F_{19} = 4181 = 37 \times 113$

Also see

 * Prime Fibonacci Number has Prime Index except for 3