Let a and n be positive integers such that f(x) = xn + a is irreducible over the integers. A conjecture made by Bouniakowsky [4] in 1857 would imply that there exist infinitely many integers x such that f(x) is prime. An even stronger conjecture of Bateman and Horn [1, 2] would imply that
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00006465/resource/name/S0008414X00006465_eqn1.gif?pub-status=live)
where π(x;f) is the number of integers m with 0 ≧ m ≧ x for which f(m) is prime, and
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00006465/resource/name/S0008414X00006465_eqn2.gif?pub-status=live)
where w(p) is the number of solutions of the congruence
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00006465/resource/name/S0008414X00006465_eqn3.gif?pub-status=live)
Except for the trivial case n = 1, neither of these conjectures has ever been resolved.