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 .
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).
Bouniakowsky, V. "Nouveaux théorèmes relatifs à la distinction des nombres premiers et à la de composition
des entiers en facteurs." Sc. Math. Phys.6, 305-329, 1857.Dickson,
L. E. History
of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York:
Dover, pp. 332-333, 2005.Ruppert, W. M. "Reducibility
of Polynomials
Modulo ."
5 Aug 1998. http://arxiv.org/abs/math.NT/9808021.Sloane,
N. J. A. Sequence A122131 in "The
On-Line Encyclopedia of Integer Sequences."