Twin primes of form $2^n+3$ and $2^n+5$ How to prove that $2^n+3$ and $2^n+5$ are both prime for only finitely many integers $n$?
And how to prove that there are infinitely many primes of the form $2^n+3$ and $2^m+5$


And how to prove that there are infinitely many primes of the form $2^n+3$ 

First prove that there are infinitely Mersenne primes, then...
This is something that can be predicted, not proved, in the current state of number theory.
The "probability" of prime numbers near $2^n$ is close to
$1/\log (2^n) = \frac{1}{n \log 2}$ and the "expected number" of primes of the form $2^m+3$ in $[1,n]$ is $O(\log \log n)$.
