Is the language { | p and n are natural numbers and there’s no prime number in [p,p+n]} belongs to NP class?

I was wondering if the following language belongs to NP class and if its complimentary belongs to NP class:

\begin{align} C=\left\{\langle p,n\rangle\mid\right.&\ \left. p \text{ and $ n$ are natural numbers}\right.\ &\left.\text{ and there’s no prime number in the range}\left[p,p+n\right]\right\} \end{align}

I am not sure, but here’s what I think: for each word $ \langle p,n\rangle \in C$ we know that the word belongs to C because there exists a primal certificate – an nontrivial divisor to any of the numbers between $ [p,p+n]$ , though I am not really sure it is in NP.

regarding the complement: I think it is in NP because the compliment compositeness can be decided by guessing a factor nondeterministically. But again I am not so sure about it and I don’t know how to correctly prove and show it.

Would really appreciate your input on that as I am quite unsure and also checked textbooks and internet (and this site) about it.