Define a Bouniakowsky polynomial as an irreducible polynomial f(x) with integer coefficients, degree >1, and GCD(f(1), f(2), ...) = 1. The Bouniakowsky conjecture states that f(x) is prime for an infinite number of integers x. As an example of the greatest common divisor caveat, the polynomial 3x^2 - x + 2 is irreducible, but always divisible by 2. Irreducible degree 1 polynomials (a x + b) always generate an infinite number of primes by Dirichlet's theorem. The existence of a Bouniakowsky polynomial that can produce an infinitude of primes is undetermined. The weaker fifth Hardy-Littlewood conjecture asserts that a^2 + 1 is prime for an infinite number of integers a>1.

Dirichlet's theorem | Hardy-Littlewood conjectures | prime-generating polynomial