There is no non-constant f ∈ ℤ[X]: f(x) is prime for every integer x

Suppose by contradiction that there exists a non-constant polynomial f with coefficients in ℤ such that f(x) is prime for every integer x. Then there exists a polynomial g ∈ ℤ[X] such that f(x)=xg(x)+p for every integer x where p is prime. Then we have for every integer x that:

f(xp)=xpg(xp)+p=p(xg(xp)+1)

it follows that for every integer x p divides f(xp), which is prime, so f(xp)=p. It follows that the non-zero polynomial q(x)=f(x)-p has infinite zeros in ℤ from which q is the zero polynomial and that is f(x)=p for every integer x, so f is constant, absurd. □