Set of prime divisors of values of nonconstant polynomial with integer coefficients is infinite

Statement

Suppose  is a nonconstant polynomial. Consider the set:

.

Then, there are infinitely many primes  such that  divides  for some natural number .

Proof

Given: A nonconstant polynomial .

To prove: The set of prime divisors of ,  varying over positive integers, is infinite.

Proof:

1. We can assume that  is primitive and irreducible over . In particular, we can assume that the gcd of the coefficients of  is , and that  has a nonzero constant term: If  doesn't already satisfy these conditions, pick an irreducible primitive factor polynomial of .
2. Suppose there are only finitely many primes  for which we can find a  such that  divides . Let  be these primes. Let  and let  be any natural number. Consider . This can be written as  times a polynomial in , with the latter polynomial having constant term . In particular, the latter term cannot be a multiple of any , and hence must be . However, this cannot be true for every natural number  unless  is a constant polynomial.