Is there a specific problem that is in both NP and co-NP but not in P? A problem is in NP if a correct answer to it can be verified to be so in polynomial time.  

A problem is in co-NP if an incorrect answer to it can be verified to be so in polynomial time.  

P is a subset of the intersection of the sets NP and co-NP.

My question is: what is a specific problem that is in the intersection of co-NP &amp; NP, but is not in P?
An answer to your question would be a problem $p$ which is in $\mathop{NP}\cap\mathop{coNP}$, and thus in $\mathop{NP}$, but not in $P$. Existence of such a problem would easily imply $P\neq\mathop{NP}$, which is a seriously open problem to this date. That is to say, no one can answer your question yet.