Is the language { | p and n are natural numbers and there's no prime number in [p,p+n]} belongs to NP class?
Note that there's always a prime between $p$ and $2p$ for any $p$ thus we can eliminate $n$ as a complexity parameter, as for any non-trivial  instance we have $n &lt; p$.

The complement (there exists a prime in range $[p, p+n]$) is quite easy  (due to the shoulders of giants). There are various well-known polynomial size certificates of primality of $q$ you could return along with some $q \in [p, p+n]$ if that prime $q$ exists. Thus your language is in co-NP.

However the question of whether it is in NP or not seems very hard to me. Our input size is $b$ such that $2^b \approx p$. You need the existence of a $O(b^c)$ sized certificate that in the worst case asserts the compositeness of an exponentially large ($|[p, p+n]| \approx |[p, 2p]| \approx p \approx 2^b$) series of consecutive integers. From my intuition about number theory, finding such a certificate would be a major result, but disproving its existence would be as well. But perhaps someone with more number theoretic knowledge than me can pitch in if either of those turn out to be easy.