## Conjecture on Primality of Wagstaff Numbers

This question is related to my previous question .

Definition :

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

$W_p=\frac{2^p+1}{3}$ , where $p$ is a prime number .

Definition :

Let's define starting seed $S$ as :

$S = 3 ~\text{ if }~ p\equiv 1 \pmod 4$

$S=11 ~\text{ if }~ p\equiv 1 \pmod 6$

$S=27 ~\text{ if }~ p\equiv 11 \pmod {12} \text{ and } p \equiv 1,9 \pmod {10}$

$S=33 ~\text{ if }~ p\equiv 11 \pmod {12} \text { and } p \equiv 3,7 \pmod {10}$

Definition :

Let's define sequence $S_i$ as :

$S_i=S^4_{i-1}-4 \cdot S^2_{i-1}+2 ~\text{ with }~ S_0=S$

Conjecture :

$W_p ~; (p>3) ~\text{ is a prime iff }~ S_{\frac{p-1}{2}} \equiv S \pmod {W_p}$

I checked statement for following Wagstaff primes :

$W_5 , W_7 , W_{11} ,W_{13} , W_{17} , W_{19} , W_{23} , W_{31} , W_{43} ,W_{61} , W_{79} , W_{101} ,W_{127} , W_{167} , W_{191} , W_{199} ,W_{313} , W_{347} ,$ $W_{701} , W_{1709} ,W_{2617} , W_{3539} , W_{5807} , W_{10501} ,W_{10691} , W_{11279} ,W_{12391} , W_{14479} ,W_{42737} ,W_{83339} ,W_{95369} ,$ $W_{127031} , W_{138937} , W_{141079} , W_{267017} , W_{269987} ,W_{374321}$

According to this criteria probable prime $W_{117239}$ is a composite number .

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

Question : Are there similar primality criteria for Wagstaff numbers in the literature ?

P.S.

According to this Wikipedia article the fastest algorithm for proving the primality of Wagstaff numbers is ECCP .

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I think that various people by now have tried to explain to you that primality criteria of this kind have a chance of being proved if and only if the full factorization of N-1 or N+1 is known. For other numbers, such as yours, these tests are simple primality tests similar to tests based on Fermat's theorem or on Lucas numbers. So please . . . ! – Franz Lemmermeyer Mar 30 2012 at 9:40