Bunyakovsky conjecture

Statement

The conjecture (not yet proved or disproved) states the following: suppose is irreducible, i.e., is an irreducible polynomial of degree two or higher with integer coefficients. Consider the set:

Then, one of these two cases must hold:

1. The greatest common divisor of all the elements of is greater than 1, i.e., all elements of have a nontrivial common factor.
2. contains infinitely many prime numbers.

Note that the first case occurs if and only if the polynomial can be written as (a positive integer greater than 1) times (an integer-valued polynomial).

Related facts and conjectures

Related known facts

• Dirichlet's theorem on primes in arithmetic progressions is the analogous statement for polynomials of degree one.
• Cohn's irreducibility criterion is a converse of sorts, which says that a polynomial with nonnegative coefficients takes a prime value at any natural number greater than all coefficients, then the polynomial is irreducible.

Related conjectures

• Bateman-Horn conjecture is a stronger conjecture that also makes assertions about the frequency of primes in the set of values taken by a polynomial.