Mersenne Primes p which are in a set of twin primes is finite?
I want to bring up the of the Mersenne Prime 2^n1 being the second prime p+2 in a twin prime pair {p, p+2} are there finitely many Mersenne Primes which hold this condition (this is the same as primes p such that 2^p1 and 2^p3 are prime).
First off 2^n1 and 2^n+1 cannot both be prime for n > 2, therefore we only focus on 2^n1 and 2^n3 both being primes.
Second, if 2^n1 and 2^n3 are both prime, n must be prime because if n is composite = ab, then 2^n1 = (2^a1)*(1 + 2^a + 2^(2*a) + 2^(3*a) .... + 2^(b*aa)
Third, if 2^n1 and 2^n3 are both prime, n = 1 (mod 4), because if n = 3 (mod 4), 2^n3 = 0 (mod 5) cannot be a prime. This follows from 2^(4*n+3) = 3 (mod 5)  3 = 0 (mod 5).
The only known exponents for which 2^n1 and 2^n3 are 3 and 5 (up to the Same Limit the Mersenne Numbers were tested). This is conjectured to be finite unless anyone brings up an arguments as to maybe why not.
Are there any more restrictions to this? Thanks for help.
