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 $2^m+5$

Settling the twin prime part of the question seems to go as follows:
1) If $2\mid n$, then $3\mid 2^n+5$, so it suffices to study odd values of $n$.
2) If $n\equiv 1\pmod 4$, then $5\mid 2^n+3$, so it suffices to study the case $n\equiv 3\pmod 4$.
3) If $n\equiv 1\pmod 3$, then $7\mid 2^n+5$. If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$, so for there to be infinitely many twin primes of this form we must have $3\mid n$.
Combinining items 2 and 3 above, we see that it suffices to exclude the possibility of infinitely many twin primes of this form, when $n\equiv 3\pmod {12}$. But...
4) If $n\equiv 3\pmod {12}$, then $13\mid 2^n+5$.
Looks like $(5,7)$ and $(11,13)$ are all.
