OPEN

Let $C>1$. Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?

Originally asked to Erdős by Kalmár. Erdős believed the answer is yes. Romanoff

[Ro34] proved that the answer is yes if $C$ is an integer.