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

From Number

## 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

- Every integer is a quadratic residue for infinitely many primes
- Every integer is a m-adic residue for infinitely many primes
- There are infinitely many primes that are one modulo any modulus
- Set of prime divisors of values of nonconstant integer-valued polynomial is infinite: An integer-valued polynomial is a polynomial with rational coefficients that sends integers to integers. We can generalize the statement of the result to nonconstant integer-valued polynomials.

## Proof

**Given**: A nonconstant polynomial .

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

**Proof**:

- 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 .
- 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.