Abstract. Hardy and Littlewood’s Conjecture F implies that the asymptotic density of prime values of the polynomials fA(x) =x 2 + x + A, A ∈ Z, is related to the discriminant ∆ = 1 − 4A of fA(x) viaaquantityC(∆). The larger C(∆) is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant ∆. A technique of Bach allows one to estimate C(∆) accurately for any ∆ < 0, given the class number of the imaginary quadratic order with discriminant ∆, and for any ∆> 0given the class number and regulator of the real quadratic order with discriminant ∆. The Manitoba Scalable Sieve Unit (MSSU) has shown us how to rapidly generate many discriminants ∆ for which C(∆) is potentially large, and new methods for evaluating class numbers and regulators of quadratic orders allow us to compute accurate estimates of C(∆) efficiently, even for values of ∆ 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(∆) is larger than any previously known examples. 1.

? ) E. Teske 1 and H.C. Williams ??2 1 Technische Universitat Darmstadt Institut fur Theoretische Informatik Alexanderstrae 10, 64283 Darmstadt Germany 2 University of Manitoba Dept. of Computer Science Winnipeg, MB Canada R3T 2N2 Abstract. Let p be a prime congruent to 1 modulo 4, n p the Legendre symbol and S(k) = P p 1 n=1 n k n p . The problem of nding a prime p such that S(3) > 0 was one of the motivating forces behind the development of several of Shanks' ideas for computing in algebraic number elds, although neither he nor D. H. and Emma Lehmer were ever successful in nding such a p. In this extended abstract we summarize some techniques which were successful in producing, for each k such that 3 k 2000, a value for p such that S(k) > 0. 1 Introduction Let d denote a fundamental discriminant of an imaginary quadratic eld IK = Q( p d ) and let h(d) denote the class number of IK. Let p be a prime ( 3(mod 4)), n p the Legendre symbol and S...