# Bunyakovsky conjecture

From Number

## Contents

## 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:

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