How to prove following statement :
Let's define an infinite sequence of positive integers :
$ a_i=\cos(4^{i} \cdot \arccos(4)) ; i=1,2,3......$
then for $n \geq 2 , F_n$ is prime if and only if :
$a_{2^{n-1}-1} \equiv 0 \pmod {F_n}$
For example :
$ a_1 \equiv 0 \pmod {F_2}$
$ a_3 \equiv 0 \pmod {F_3}$
$ a_7 \equiv 0 \pmod {F_4}$