Here is what I observed :
Let's put the number $W_p = \frac{2^p+1}{3}$ where $p$ is prime and $W_p$ is a Wagstaff number.
Let's put the formula $\omega_p = (\frac{1}{2} (1 + i \sqrt7))^{(\frac{2^{p - 1} - 1}{3})} + (\frac{1}{2} (1 - i \sqrt7))^{(\frac{2^{p - 1} - 1}{3})}$ when $p$ is a prime number $>3$
Then it seems than $W_p$ divides $\omega_p$ only if $W_p$ is prime.
For example :
- $W_5 = 11$ divides $\omega_5 = 11$
- $W_7 = 43$ divides $\omega_7 = 2795$
- $W_{11} = 683$ divides $\omega_{11} = -2740121063461600091886597704982460332288405576168469$
- $W_{29}$ doesn't divide $\omega_{29}$ the first non-Wagstaff prime.
I've checked until $p = 31$ and it seems it's true.
So I have two questions :
Do you think this is possible to make a primality test like Lucas-Lehmer test for this ? I tried to find a sequence but I didn't find it for the moment.
And if yes, is there a way to prove it ?