Mersenne Numbers & Primes

In the statements below p is an odd prime.
All 2^p-1 have the form 8k+7
They also have the form pk+1, This is from the Little Fermat Theorem
where 2^p-2 is 0 mod p therefore by adding 1 to both sides we get
2^p-1 is 1 mod p.
This may be a clue to the factors of 2^p-1 that are composite.
My guess is that pk+1 is a factor, k < (2^p-1)/p
Here is a look at the Mersenne Composites.
Another obvious fact is that 2^(p-1)-1 is 0 mod p, E.G. 2²⁸-1 has a factor of 29.
There is a test called the Lucas-Lehmer Test to prove primality for Mersenne Numbers.
Here is a another tidbit If 2^p-1 is prime then 2^p+1 is divisible by 3. 2^2p-1 can be factored into (2^p-1)(2^p+1)
since 2^2p-1 is also factorable by 2²-1 because the exponent is composite
since 2^p-1 is prime and 2^2p-1 factors to (2^p-1)(2^p+1)
2²-1 or 3 must divide 2^p+1