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I asked this question on mathstackexchange but didn't get any answer .

Definition :

Let $W_p$ be a Wagstaff number of the form :

$W_p=\frac{2^p+1}{3}$ , with $p\equiv 1 \pmod 4$

Next , define sequence $S_i$ as :

$S_i =8S^4_{i-1}-8S^2_{i-1}+1 $ , with $ S_0=\frac{3}{2} $

How to prove following statement :

Conjecture :

$W_p$ is a prime iff $S_{\frac{p-1}{2}} \equiv \frac{3}{2} \pmod {W_p}$

I checked statement for following Wagstaff primes :

$W_5 , W_{13} , W_{17} , W_{61} , W_{101} , W_{313} , W_{701} , W_{1709} , W_{2617} , W_{10501} , W_{42737} ,W_{95369} , W_{138937} ,W_{267017}$

Also , for $~p < 15000~$ there is no composite $W_p$ that satisfies relation from conjecture .

P.S.

I am interested in hints (not full solution) .

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Let $Y_0 = 3$ and $Y_{i+1} = Y_i^2-2$. Then your $S_i = \frac{1}{2} Y_{2i}$. So your condition would be that $Y_{p-1}\equiv Y_0 \pmod{W_p}$. This is nearly the same as (and I'd say equivalent to) the second conjecture posed here on mersenneforum.org. See the link for some discussion, partial results and variations of the test.

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Look at this paper: http://trex58.files.wordpress.com/2009/01/wagstaffandfermat.pdf for a proof of the first part of the 2nd conjecture (Wagstaff numbers) in Mersenne forum and look at my paper: http://tony.reix.free.fr/Mersenne/ConjectureLLTCyclesMersenne.pdf for a proof of the first part of the 1rst conjecture on same thread of Mersenne forum.

Proving the converse seems VERY difficult, if provable.

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