# Primality criterion for generalized Fermat numbers similar to the LLT ?

I asked this question on mathstackexchange but didn't get any answer .

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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