Bouniakowsky Conjecture

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 .

Various prime-generating polynomials are known, but none of these always generates a prime (Legendre).

Worse yet, it is unknown if a general Bouniakowsky polynomial will always produce at least 1 prime number. For example, produces no primes until , 764400, 933660, ... (OEIS A122131).