Fibonacci Number with Prime Index is not necessarily Prime

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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:

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


$\blacksquare$


Also see


Sources