text: # 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) .
  
  -
explanation: The document presents a mathematical exploration of a primality criterion for generalized Fermat numbers, similar to the Lucas-Lehmer test (LLT). It includes definitions, examples, and conjectures related to the divisibility of certain sequences by generalized Fermat numbers. The document also mentions that the author has posted a similar question on MathStackExchange but did not receive an answer. The text ends with a request for hints rather than a full solution.
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distilabel_metadata.content: # 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) .
  
  -
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document presents a mathematical exploration of a primality criterion for generalized Fermat numbers, similar to the Lucas-Lehmer test (LLT). It includes definitions, examples, and conjectures related to the divisibility of certain sequences by generalized Fermat numbers. The document also mentions that the author has posted a similar question on MathStackExchange but did not receive an answer. The text ends with a request for hints rather than a full solution.
  
  Label: question-and-answer