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# Polynomials of quadratic type producing strings of primes

Published:1997-06-01
Printed: Jun 1997
• R. A. Mollin
• B. Goddard
• S. Coupland
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## Abstract

The primary purpose of this paper is to provide necessary and sufficient conditions for certain quadratic polynomials of negative discriminant (which we call Euler-Rabinowitsch type), to produce consecutive prime values for an initial range of input values less than a Minkowski bound. This not only generalizes the classical work of Frobenius, the later developments by Hendy, and the generalizations by others, but also concludes the line of reasoning by providing a complete list of all such prime-producing polynomials, under the assumption of the generalized Riemann hypothesis ($\GRH$). We demonstrate how this prime-production phenomenon is related to the exponent of the class group of the underlying complex quadratic field. Numerous examples, and a remaining conjecture, are also given.
 MSC Classifications: 11R11 - Quadratic extensions 11R09 - Polynomials (irreducibility, etc.) 11R29 - Class numbers, class groups, discriminants