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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 ) $

How to prove following statements :

Example 1 :

Let's define sequence $S_i $ as :

$ S_i = T_{78} (S_{i-1} ) \text{ with } S_{0} = 6 $

I have found that :

$ F_1(156) \mid S_2 , ~ F_4(156) \mid S_{30} , ~ F_5(156) \mid S_{62} $

Also , no composite $ F_n(156) $ up to $n= 10 $ divides corresponding $S_i $ .

Conjecture :

$F_{n}(156) , (n \geq 1) \text{ is a prime iff } F_n(156) \mid S_{2^{n+1}-2} $

Example 2 :

Let's define sequence $S_i $ as :

$ S_i = T_{18} (S_{i-1} ) \text{ with } S_{0} = 8 $

I have found that :

$ F_2(288) \mid S_{17} , ~ F_3(288) \mid S_{37} , ~ F_4(288) \mid S_{77} $

Also , no composite $F_{n}(288) $ up to $ n=10 $ divides corresponding $S_i $ .

Conjecture :

$F_{n}(288) , (n \geq 1) \text{ is a prime iff } F_n(288) \mid S_{5 \cdot 2^n-3} $

Example 3 :

Let's define sequence $S_i $ as :

$S_i = T_{2222}(S_{i-1}) \text{ with } S_{0}=4 $

I have found that :

$ F_1(4444) \mid S_{2} , ~ F_2(4444) \mid S_{6} , ~ F_4(4444) \mid S_{30} $

Also , no composite $F_{n}(4444) $ up to $n=10 $ divides corresponding $S_i $ .

Conjecture :

$F_{n}(4444) , (n \geq 1 ) \text{ is a prime iff } F_n(4444) \mid S_{2^{n+1}-2} $

P.S.

I am interested in hints (not full solution) .

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