I want to know if it is possible to find a formula that gives all the prime numbers? or can one find the distribution of prime numbers?
I know that there is a set of ongoing research on prime numbers. I know that there is already lot of results in prime numbers, the gap between primes, a lot of theorems starting from Euclid's one. My question is simple: Is it possible for someone to find explicit formula that gives all prime numbers (explicit function $f(x)$ is a prime for any natural number $x$)? or that gives the distribution of them? If it is not possible. Suppose that someone prove that factoring is an easy problem. Still not possible?
I mean, is there a theorem, logic, something that says that it is impossible to get all the primes or that all primes can be calculated based on condition $1$, $2$, $\dotsc$?
I hope that I make my question clear. I hope that it is a valid question.
Thank you very much for your help. Any reference or ideas would be appreciated.