Bateman-Horn conjecture

In number theory, the Bateman-Horn conjecture is 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.

It provides a conjectured density for the condition that all of a set of polynomials evaluated at a positive integer give a prime number. The set of polynomials $f_1, \cdots f_m$ are m distinct, irreducible polynomials with integer coefficients, such that the leading coefficients are positive and such that if f is the product of all the polynomials f_i, then there does not exist a prime number p dividing f(n) for every positive integer n. If P(x) is the number of positive integers less than x such that all of the polynomials evaluate to a prime, then the conjecture is

$P(x) \sim \frac{C}{D} \int_2^x \frac{dz}{(\ln z)^m},\,$

where C is the product over primes p

$C = \prod_p \frac{1-N(p)/p}{(1-1/p)^m}$

with $N (p)$ the number of mod p solutions to $f(n) \equiv 0 \pmod p$ where f is the product of the polynomials f_i, and D is the product of the degrees of the polynomials, or in other words the degree of f.

References

Bateman, P. T. and Horn, R. A., A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation 16 (1962), pp 363-367

Categories: Number theory | Conjectures