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 .
