Proof by reductio ad impossibile.
Suppose that there is exist such polynomial .
Let now such that , when is prime.
Let now be , and let us look at:
is polynomial with integer coefficients.
So, , but is generates primes only, therefor for all integer .
Now, from the fact that polynomial can't have the same value more than times, we get the contradiction!
Here some interesting polynomials:
when then is prime...