# Bateman-Horn conjecture

A conjecture on the asymptotic behaviour of a polynomial satisfying the Bunyakovskii condition (cf. also Bunyakovskii conjecture).

Let $f _ {1} ( x ) \dots f _ {r} ( x )$ be polynomials (cf. Polynomial) with integer coefficients, of degrees $d _ {1} \dots d _ {r} \geq 1$, irreducible (cf. Irreducible polynomial), and with positive leading coefficients. Let

$$f = f _ {1} \dots f _ {r}$$

be their product.

V. Bunyakovskii considered the case $r = 1$ and asked whether $f ( n )$ could represent infinitely many prime numbers as $n$ ranges over the positive integers. An obvious necessary condition is that all coefficients of $f$ be relatively prime. However, that is not sufficient. He conjectured that, in addition, the following Bunyakovskii condition is sufficient: there is no prime number $p$ dividing all the values $f ( n )$ for the positive integers $n$( cf. Bunyakovskii conjecture).

Assuming the Bunyakovskii condition, let

$$C ( f ) = \prod _ {p \textrm{ a prime } } \left ( 1 - { \frac{1}{p} } \right ) ^ {- r } \left ( 1 - { \frac{N _ {f} ( p ) }{p} } \right ) ,$$

where $N _ {f} ( p )$ is the number of solutions of the congruence equation $f ( n ) \equiv0 ( { \mathop{\rm mod} } p )$( for $p$ prime). The Bateman–Horn conjecture asserts that

$$\pi _ {f} ( x ) \sim { \frac{C ( f ) }{d _ {1} \dots d _ {r} } } \int\limits _ { 2 } ^ { x } { { \frac{1}{( { \mathop{\rm log} } t ) ^ {r} } } } {dt } ,$$

where $\pi _ {f} ( x )$ is the number of positive integers $n \leq x$ such that all $f _ {1} ( n ) \dots f _ {r} ( n )$ are prime.

This formula gives the density of primes in an arithmetic progression (cf. Dirichlet theorem), using the polynomial $f ( x ) = ax + b$. After some computations, it gives the asymptotic behaviour conjectured by G.H. Hardy and J.E. Littlewood for the number of primes representable by the polynomial $x ^ {2} + 1$. It also gives the Hardy–Littlewood conjecture for the behaviour of the number of twin primes, by applying the formula to the polynomials $x$ and $x + 2$( cf. also Twins). Similarly, it implies many other conjectures of Hardy and Littlewood stated in [a2].