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}
could you please check if my reasoning is okay to deduce NP? (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.
Edit: for the sake of solving the problem, due to xskxzr's comment, let's assume p and n are represented by binary, as there's a difference according to his comment between p and n being represented in unary and binary(this is also quite interesting).