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 "n"²+1; it is also a strengthening of Schinzel's hypothesis H.

It provides a conjectured density for the positive integers at which a given set of polynomials all have prime values. The set of polynomials $f\_1,\; dots,\; f\_m$ are "m" distinct, irreducible polynomials with integer coefficients, such that that the product "f" of all the polynomials "f"_i has Bunyakovsky's property: no prime number "p" divides "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:$P(x)\; sim\; frac\{C\}\{D\}\; int\_2^x\; frac\{dt\}\{(log\; t)^m\},,$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.

Often this conjecture assumes the polynomials $f\_i$ have positive leading coefficient. This is an irrelevant condition if one allows negative primes (whichis reasonable if you try to formulate the conjecture beyond the classical case of the integers), but at the same time it is easy to just negate the polynomials if necessary to reduce to the case where the leading coefficients are positive.

Bunyakovsky's property implies

:"N"("p") < "p"

for all primes "p", so each factor in the infinite product "C" is positive. Intuitively one then naturally expects that the constant "C" is itself positive, and with some work this can be proved. (Work is needed since some infinite products of positive numbers equal zero.)

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