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) .