Does there exist an explicit formula that expresses $p_{n+1}-p_{n}$ in terms of $p_n$, the $n$th prime number?

Question

Given the $n$th prime number, is it possible to express explicitly a function $\phi:P \mapsto \mathbb{N}$, where $P$ is the set of all prime numbers, such that $\phi(p_n)$ gives the prime gap after $p_n$?

Answer 1

No, I don't think so. Because otherwise we would have a function which gives us all the prime numbers. It is easily seen, that we would have $$p_{n + 1} = p_n + \phi(p_n)$$ and inductively we would get every prime number.