I searched for natural numbers $n$, where $2^n-n$ and $2^n+n$ are both prime for the range of $n \le 10^5$ on PARI/GP and found that 3, 9 are the only solutions in this range.
Note that since $2^n-n$, $2^n$, $2^n+n$ are in arithmetic progression and $2^n$ is not divisible by 3, $n$ must be a multiple of 3.
Questions:
$(1)$ Are $3$, $9$ the only natural numbers $n$ for which both $2^n-n$ and $2^n+n$ are prime?
$(2)$ If not, then can you prove or disprove that there are finite natural numbers of this type?