I know that it is known that for every integer $n>1$ there is always a prime $p$ such that $n<p<2n$ (Bertrand´s postulate).
Also, I guess that it is still not known whether there is at least one prime between $n^2$ and $(n+1)^2$, for every positive integer $n$ (Legendre´s conjecture).
What about this:
Is there always at least one prime in the closed interval $[2^n,2^n+n]$ for every positive integer $n$?
I just checked for the first few $n$ by heart and found no counterexample although sometimes primes are at the endpoints of these closed intervals. Maybe first counterexample, if it exists, is not far away, but somebody will, I hope, check that.