Definition 1 :

Let F_n(b) be a generalized Fermat number of the form :

F_n(b) = b^{2^n}+1 , where b is a positive even integer .

Definition 2 :

Let T_n(S_{i-1}) be a Chebyshev polynomial of the first kind , i.e.

T_n(S_{i-1}) =  2^{-1} \cdot \left(\left(S_{i-1}+\sqrt{S^2_{i-1}-1}\right)^n+\left(S_{i-1}-\sqrt{S^2_{i-1}-1}\right)^n\right )

Definition 3 :

Let's define sequence S_i as :

 S_i = T_{146} (S_{i-1} ) \text{ with }  S_{0} = 4

I have found that :

 F_1(2336) \mid S_8  , ~ F_2(2336) \mid S_{18} , ~ F_6(2336) \mid S_{318}

Also , no composite  F_n(2336) up to n= 10 divides corresponding  S_i .

How to prove following statement :

Conjecture :

F_{n}(2336) , (n \geq 1)   \text{ is a prime iff }  F_n(2336) \mid S_{5\cdot2^{n}-2}