Prime Values of Polynomials and Irreducibility Testing

In 1857 Bouniakowsky [6] made a conjecture concerning prime values of polynomials that would, for instance, imply that x + 1 is prime for infinitely many integers x. Let ƒ (x) be a polynomial with integer coefficients and define the fixed divisor of ƒ, written d(ƒ), as the largest integer d such that d divides f(x) for all integers x. Bouniakowsky conjectured that if f(x) is nonconstant and irreducible over the integers, then there exist infinitely many integers x such that f(x)/d(f) is a prime. An even stronger conjecture of Bateman and Horn [3, 4] would imply that if ƒ (x) is a nonconstant irreducible polynomial of degree n, with d(f) = 1, then