Батеман-Горн болжамы және полиномдардың иррационалдық мәндерінің жиілігі

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 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.

If the system of polynomials consists of the single polynomial ƒ1(x) = x, then the values n for which ƒ1(n) is prime are themselves the prime numbers, and the conjecture becomes a restatement of the prime number theorem. If the system of polynomials consists of the two polynomials ƒ1(x) = x and ƒ2(x) = x + 2, then the values of n for which both ƒ1(n) and ƒ2(n) are prime are just the smaller of the two primes in every pair of twin primes. In this case, the Bateman–Horn conjecture reduces to the Hardy–Littlewood conjecture on the density of twin primes, according to which the number of twin prime pairs less than x is

When the integers are replaced by the polynomial ring F[u] for a finite field F, one can ask how often a finite set of polynomials fi(x) in F[u][x] simultaneously takes irreducible values in F[u] when we substitute for x elements of F[u]. Well known analogies between integers and F[u] suggest an analogue of the Bateman–Horn conjecture over F[u], but the analogue is wrong. For example, data suggest that the polynomial

in F3[u][x] takes (asymptotically) the expected number of irreducible values when x runs over polynomials in F3[u] of odd degree, but it appears to take (asymptotically) twice as many irreducible values as expected when x runs over polynomials of degree that is 2 mod 4, while it (provably) takes no irreducible values at all when x runs over nonconstant polynomials with degree that is a multiple of 4. An analogue of the Bateman–Horn conjecture over F[u] which fits numerical data uses an additional factor in the asymptotics which depends on the value of d mod 4, where d is the degree of the polynomials in F[u] over which x is sampled.