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Hardy and Littlewood's Conjecture F implies that the asymptotic density of prime values of the polynomials fA(x) = x2 + x + A, A ∈ Z, is related to the discriminant Δ = 1 - 4A of fA(x) via a quantity C(Δ). The larger C(Δ) is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant Δ. A technique of Bach allows one to estimate C(Δ) accurately for any Δ 0 given the class number and regulator of the real quadratic order with discriminant Δ. The Manitoba Scalable Sieve Unit (MSSU) has shown us how to rapidly generate many discriminants Δ for which C(Δ) is potentially large, and new methods for evaluating class numbers and regulators of quadratic orders allow us co compute accurate estimates of C(Δ) efficiently, even for values of Δ with as many as 70 decimal digits. Using these methods, we were able to find a number of discriminants for which, under the assumption of the Extended Riemann Hypothesis, C(Δ) is larger than any previously known examples.

New quadratic polynomials with high densities of prime values