Bateman–Horn conjecture
In number theory, the Bateman–Horn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A Horn, of The University of Utah, who proposed it in 1962. It provides a vast generalization of such conjectures as the Hardy and Littlewood conjecture on the density of twin primes or their conjecture on primes of the form n2 + 1; it is also a strengthening of Schinzel's hypothesis H.
The Bateman–Horn conjecture provides a conjectured density for the positive integers at which a given set of polynomials all have prime values. For a set of m distinct irreducible polynomials ƒ1, ..., ƒm with integer coefficients, an obvious necessary condition for the polynomials to simultaneously generate prime values infinitely often is that they satisfy Bunyakovsky's property, that there does not exist a prime number p that divides their product f(n) for every positive integer n. For, if there were such a prime p, having all values of the polynomials simultaneously prime for a given n would imply that at least one of them must be equal to p, which can only happen for finitely many values of n.
An integer n is prime-generating for the given system of polynomials if every polynomial ƒi(n) produces a prime number when given n as its argument. If P(x) is the number of prime-generating integers among the positive integers less than x, then the Bateman–Horn conjecture states that
where D is the product of the degrees of the polynomials and where C is the product over primes p
with the number of solutions to
...
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