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

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Statement

Suppose is a nonconstant polynomial. Consider the set:

.

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

Related facts

Stronger facts

Applications

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.