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You probably already know the Lucas-Lehmer-Test (LLT) used for proving that a Mersenne number is prime or composite. (See: http://mathworld.wolfram.com/Lucas-LehmerTest.html" [Broken]).

The LLT is based on the properties of the Tree built by [tex]x^2-2[/tex] modulo a Mersenne number.

Now, here is a conjecture (checked up to M26 = [tex]M_{23209}[/tex]) based on the properties of the Cycles of length (q-1) built by [tex]x^2-2[/tex] modulo a Mersenne number.

[tex]\large M_q=2^q-1 \text{ is prime } \Longleftrightarrow \ S_{q-1} \equiv S_0 \ \pmod{M_q} \text{ , where: } S_0=3^2+1/3^2 , \ S_{i+1}=S_i^2-2 \ .[/tex]

How can it be proved ?

Tony

(far from Internet till 8th of May)

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# Another (candidate) test of primality of Mersenne number

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