The wiki page on Mersenne Primes gives 8 theorems about Mersenne primes. My question relates to number 4. and 7.:
4.If $p$ is an odd prime, then any prime $q$ that divides $2^p-1$ must be congruent to $\pm 1 (\bmod 8)$.
7.If $p$ and $2p+1$ are both prime (meaning that $p$ is a Sophie Germain prime), and $p$ is congruent to $3 (\bmod 4)$, then $2p+1$ divides $2^p − 1$.
I have 3 questions:
Isn't this equivalent, since from 7. we get $2p+1=8n +7$ which is $- 1 (\bmod 8)$? Ok, I see that 7. is more stringent, since 4. also allows $1$ as divisor. If this is answer to 1., my next question is
How does the property of $p$ and $2p+1$ both prime force that $2p+1$ divides $2^p − 1$?
Is this extendable? Can a Cunningham chain help to find more divisors of $2^p − 1$?