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Mathematics of Computation

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New quadratic polynomials with high densities of prime values

Authors: Michael J. Jacobson Jr. and Hugh C. Williams
Journal: Math. Comp. 72 (2003), 499-519
MSC (2000): Primary 11R11, 11R29, 11Y40; Secondary 11Y16
Published electronically: May 2, 2002
MathSciNet review: 1933834
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Abstract: Hardy and Littlewood's Conjecture F implies that the asymptotic density of prime values of the polynomials $f_A(x) = x^2 + x + A,$ $A \in \mathbb{Z}$, is related to the discriminant $\Delta = 1 - 4A$ of $f_A(x)$ via a quantity $C(\Delta).$ The larger $C(\Delta)$ is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant $\Delta$. A technique of Bach allows one to estimate $C(\Delta)$ accurately for any $\Delta < 0$, given the class number of the imaginary quadratic order with discriminant $\Delta$, and for any $\Delta > 0$ given the class number and regulator of the real quadratic order with discriminant $\Delta$. The Manitoba Scalable Sieve Unit (MSSU) has shown us how to rapidly generate many discriminants $\Delta$ for which $C(\Delta)$ is potentially large, and new methods for evaluating class numbers and regulators of quadratic orders allow us to compute accurate estimates of $C(\Delta)$ efficiently, even for values of $\Delta$ 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, $C(\Delta)$ is larger than any previously known examples.

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Additional Information

Michael J. Jacobson Jr.
Affiliation: Department of Computer Science, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

Hugh C. Williams
Affiliation: Department of Mathematics and Statistics, MS 360, 2500 University Drive N.W., University of Calgary, Calgary, Alberta, Canada T2N 1N4

Keywords: Prime-generating quadratic polynomial, quadratic order, class group
Received by editor(s): September 8, 1999
Received by editor(s) in revised form: February 28, 2001
Published electronically: May 2, 2002
Article copyright: © Copyright 2002 American Mathematical Society