It is easy to see that \(\textbf{NP} \cap \textbf{coNP}\) includes all problems in \(\textbf{P}\). But this class also contains some problems which are not currently known to be feasibly decidable. An important example is \(\sc{FACTORIZATION}\) (as defined in  Section 1.1).   For on the one hand, a number \(1 \lt d \leq m\) which divides \(n\) serves as a certificate for the membership of \(\langle n,m \rangle\) in \(\sc{FACTORIZATION}\). And on the other hand, in order to demonstrate the membership of \(\langle n,m \rangle\) in \(\overline{\sc{FACTORIZATION}}\), it suffices to exhibit a prime factorization of \(n\) in which no factor is less than \(m\). This follows because prime factorizations are unique (unlike, e.g., falsifying valuations in the case of \(\sc{VALIDITY}\)) and because the primality of the individual factors can be verified in polynomial time by the AKS algorithm. Hence \(\sc{FACTORIZATION}\) is in \(\textbf{NP}\) and \(\textbf{coNP}\) simultaneously.