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Polynomials of Quadratic Type Producing Strings of Primes
Published online by Cambridge University Press: 20 November 2018
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 primeproducing 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.
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- Copyright © Canadian Mathematical Society 1997