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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 .


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 See the link for some discussion, partial results and variations of the test.

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Look at this paper: for a proof of the first part of the 2nd conjecture (Wagstaff numbers) in Mersenne forum and look at my paper: 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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