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Note : This problem has no specific source

Def :

Let's define number $N$ as : $N=3\cdot 2^n+1$

Def :

Let's define starting seed $S$ as :

$S = \begin{cases} 32672, & \text{if } n\equiv 1 \pmod 4 \\ 21868, & \text{if }n\equiv 2 \pmod 4 \end{cases}$

Def :

Let's define sequence $S_i$ as :

$S_i=S^2_{i-1}-2~$ with $~S_0=S$

Conjecture :

$N ; (n > 2)~$ is a prime iff $S_{n-2} \equiv 0 \pmod N$

I have checked statement for all exponents under $100000$ from this list .

Question :

I am interested in approaches which can be used to prove ( disprove ) this conjecture .

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The best approach for disproving such a conjecture would be looking for a counterexample by testing composite numbers as well. The best approach for understanding the problem would be to familiarize yourself with the standard primality tests based on the factorizations of n+1 and n-1. But perhaps this is not an option as this would keep you from posting this question over and over again. – franz lemmermeyer May 5 '12 at 11:48

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