Are there infinitely many primes of the form $3^p-4$? My question is the same as you saw:

Are there infinitely many primes of the form $3^p-4$, when $p$ prime?

I'm a complete novice in this type of questions with few or no theoretical background. Um, so I could only try it by brute-force for all primes $p<15,000$. Then, I found something interesting: it seems not so plausible, because $p$ with $3^p-4$ prime gets rapidly rare, as $p$ increases.
The complete list when $p<15,000$ is: $\{2,3,5,31,37,41,53,73,101,557\}$. The largest integer $n<15,000$ with $3^n-4$ prime I found is $4823=7\times 13\times 53$.
It seems quite hard, so I just hope I could find some references or survey articles. At least, is there a verified infinite sequence of primes in a set $\{a^p-b^q:$ $p,q$ primes$\}$ for some fixed primes $a,b$?

It is not known in general whether or not there are infinitely many primes of the form $a^p+b$. Some results are known for $a=2$, e.g., see this MO-question. 
Even for polynomials of degree $\ge 2$ the question is not known. There we have   the so-called Bunyakovsky conjecture.
