Lucas-Lehmer test for Mersenne and Wagstaff numbers?

Question

Here is what I observed :

Let $M_p = 2^p-1$ for Mersenne numbers and $W_p = (2^p+1)/3$ for Wagstaff numbers with $p$ a prime number > $2$

Let the sequence $S_i = 6 \cdot S_{i-1}^2 + 18 \cdot S_{i-1} +12$ with $S_0 = 12$

Then $M_p$ or $W_p$ is prime if $S_{p-2} \equiv 0 \pmod{M_p}$ or $S_{p-2} \equiv 0 \pmod{W_p}$

For example let's take $17$ for both case with PARI GP :

For $M_{17}$ we get :

Mod(12, 131071)
Mod(1092, 131071)
Mod(96618, 131071)
Mod(85540, 131071)
Mod(22888, 131071)
Mod(99467, 131071)
Mod(4058, 131071)
Mod(49706, 131071)
Mod(96810, 131071)
Mod(97352, 131071)
Mod(21854, 131071)
Mod(109865, 131071)
Mod(81598, 131071)
Mod(36387, 131071)
Mod(131069, 131071)
Mod(0, 131071)

$S_{p-2} \equiv 0 \pmod{M_{17}}$ so $131071$ is prime.

For $W_{17}$ we get :

Mod(12, 43691)
Mod(1092, 43691)
Mod(9128, 43691)
Mod(43125, 43691)
Mod(33247, 43691)
Mod(2111, 43691)
Mod(37044, 43691)
Mod(33796, 43691)
Mod(37321, 43691)
Mod(31573, 43691)
Mod(3181, 43691)
Mod(39346, 43691)
Mod(36262, 43691)
Mod(3520, 43691)
Mod(43690, 43691)
Mod(0, 43691)

$S_{p-2} \equiv 0 \pmod{W_{17}}$ so $43691$ is prime.

It's better to use the original sequence $S^2-2$ for Mersenne numbers, but with the new sequence, it seems it works for both $M_p$ and $W_p$ for primality testing

Is there a way to prove this ?

I noticed than the sequence can be written as $6(S+1)(S+2)$ and when the starting value is $12$, $S = 1/6 \cdot (3^{(2^{(2 + n)})}-9)$ and interestingly the sum of the coefficients of the sequence is equal to the discriminant $36 = 6^2$.

Answer 1

If you look at the sequence in OEIS A029858 changing the index is equal to the sequence you used then you get

$$S_{p-2}=\frac{3^{2^p-1}-3}{2}$$

for Mersenne numbers

but also

$$S_{p-2}=\frac{3^{2^p-1}-3}{2}=\frac{3^{2^p+1}-27}{18}=\frac{27 ^{\frac{2^p+1}{3}}-27}{18}$$

for Wagstaff numbers

and if you apply Fermat's little theorem you get the result you observed.