We start, with the original Lucas-Lehmer test format:
$s_0=4\\ s_i=s_{i-1}^2-2 \pmod {2^p-1}$
We can note, right away, that all terms are even. Dividing out the factor of 2, we get:
$s_0=2\\ s_i=2x_{i-1}^2-1 \text{ always odd $\forall i, 0<i$} $
I know the following (From playing around quite a bit. I even have a theory of Mersenne primes thread on the mersenne forum site):
All odd numbers y=2u+1 , that aren't prime, and divide by prime z=2v+1, have $u\equiv v \pmod z $ . I'm hoping this can be used.