Are there any open problems concerning decidability? I am learning computability theory. I am just interested to know some famous problems (Formally languages) whose decidability is in question.
Given a finite automaton A over the alphabet {0,1}, does A accept the base-2 representation of at least one prime number?   This is currently not known to be either decidable or undecidable.

(By contrast, the same problem with "prime" replaced by "composite" is decidable.)