It seems there are infinitely many primes of the form $2^n + 3$.
It seems that the density is about the same density as primes below $n$ at least for small $n$.
What surprised me is I found solutions to $2^n + 3$ and $2^{n+1} + 3$ both prime.
Is the density of that about the same density as prime twins below $n$ ?
I will add some data later.
Any references to these ?
$f(n) = 2^n + 3$ is prime for
$n =$
$1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 28, 30, 55, 67$
added :
Conjecture :
If $f(k)$ and $f(k+1)$ are prime then
$$\frac{f(k+1)}{f(k)} < e$$
I could split up the question if that conjecture gets answered, well actually It might not be provable since the list is not proven infinite but a counterexample or strong argument maybe.