Bunyakovsky conjecture

Statement

The conjecture (not yet proved or disproved) states the following: suppose $p(x)\in \mathbb{Z}[x]$ is irreducible, i.e., $p$ is an irreducible polynomial of degree two or higher with integer coefficients. Consider the set:

$S=\{p(n)\mid n\in \mathbb{N}\}$

Then, one of these two cases must hold:

1. The greatest common divisor of all the elements of $S$ is greater than 1, i.e., all elements of $S$ have a nontrivial common factor.
2. $S$ 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.