Define a Bouniakowsky polynomial as an irreducible polynomial
with integer coefficients, degree , and . The Bouniakowsky conjecture states that
is prime for an infinite number of integers (Bouniakowsky 1857). As an example of the greatest
common divisor caveat, the polynomial is irreducible, but always divisible by 2.
Irreducible degree 1 polynomials () 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 is prime for an infinite number of integers .