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 :
$S_i = \begin{cases} 3/2, & i=0 \\ 8S^4_{i-1}-8S^2_{i-1}+1, & i>0 \end{cases}$
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 :
$W_5 , W_{13} , W_{17} , W_{61} , W_{101} , W_{313} , W_{701} , W_{1709} , W_{2617} , W_{10501} , W_{42737} ,W_{95369} , W_{138937} ,W_{267017}$