For example: We know for $b=12$ and $n=1$, $12^1+1=13$ is prime. A quick check with WolframAlpha shows that there no other primes of the form $12^n+1$ up to $n=1000$. Are the any other primes for this base? Also what about numbers of the form $b^n+1$ where $b$ is some fixed even integer?
My thoughts:
for $b=2^k$ $$b^n+1=2^{kn}+1$$ so if we have the primes of the form $2^n+1$, we have those of $(2^k)^n+1$ aswell.
for $b=k^2$ $$b^n+1=({k^n})^{2}+1$$ and it is an open question, as these numbers are a subset of numbers of the form $m^2+1$.
Now let $b=12$ for example, then $b$ is not a square number and no power of $2$. What can we say about this base?