Definition 1 :

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

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

Definition 2 :

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

$\displaystyle 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 $\displaystyle S_i $ as :

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

I have found that :

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

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

How to prove following statement :

Conjecture :

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