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pedja
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Primality Criteria for F_n(132)
[tex]\text{Let's define sequence}~ S_i ~\text{as :}[/tex]
[tex]S_i= T_{66}(S_{i-1})=2^{-1}\cdot \left(\left(S_{i-1}+\sqrt{S_{i-1}^2-1}\right)^{66}+\left(S_{i-1}-\sqrt{S_{i-1}^2-1}\right)^{66}\right) , ~\text{with}~ S_0=8[/tex]
[tex]\text{and define} ~F_n(132)=132^{2^n}+1[/tex]
[tex]\text{I found that :} ~F_2(132) \mid S_5 , ~ F_3(132) \mid S_{13} , ~F_5(132) \mid S_{61}[/tex]
How to prove following statement :
Conjecture :
[tex] F_n(132) ;~ (n\geq 1)~\text{ is a prime iff}~F_n(132) \mid S_{2^{n+1}-3}[/tex]
[tex]\text{Let's define sequence}~ S_i ~\text{as :}[/tex]
[tex]S_i= T_{66}(S_{i-1})=2^{-1}\cdot \left(\left(S_{i-1}+\sqrt{S_{i-1}^2-1}\right)^{66}+\left(S_{i-1}-\sqrt{S_{i-1}^2-1}\right)^{66}\right) , ~\text{with}~ S_0=8[/tex]
[tex]\text{and define} ~F_n(132)=132^{2^n}+1[/tex]
[tex]\text{I found that :} ~F_2(132) \mid S_5 , ~ F_3(132) \mid S_{13} , ~F_5(132) \mid S_{61}[/tex]
How to prove following statement :
Conjecture :
[tex] F_n(132) ;~ (n\geq 1)~\text{ is a prime iff}~F_n(132) \mid S_{2^{n+1}-3}[/tex]
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