all the polynomials to have positive leading coefficients, so that only a constant number of their values can be negative. Alternatively, one may allow negative leading coefficients but count a negative number as being prime when its absolute value is prime. It is reasonable to allow negative numbers to count as primes as a step towards formulating more general conjectures that apply to other systems of numbers than 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. Examples 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 Analogue for polynomials over a finite field 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. References .