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I've spent dozen of hours searching by my-self and dozen of hours searching on the Web. Now I need help.

Who could provide a proof for this binomial property ? I need it for another proof.

Thanks

Tony

Let: [tex]F_n=2^{2^n}+1 , n \geq 2 .[/tex]

Prove: [tex]F_n \text{ prime } \Longrightarrow

F_n \mid A_{k_n} , \text{ where } k_n=2^{3 \times 2^{n-2}-1} \text{

and } A_{k_n} = \sum_{i=0}^{k_n/2}{k_n \choose 2i}

2^i[/tex]

Examples:

[tex]n=2 , F_2=17 , k_2=4 , A_{k_2}=17[/tex]

[tex]n=3 , F_3=257 , k_3=32 , A_{k_3}=257*1409*2448769[/tex]

[tex]n=4 , F_4=65537 , k_4=2048 , A_{k_4}=\text{very big} \equiv 0 \

(\text{mod} F_4)[/tex]

[tex]n=5 , F_5=4294967297 , k_5=8388608 , A_{k_5}=\text{VERY big} \neq 0 \ (\text{mod} F_5)[/tex]

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# I need a proof for this binomial property.

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