Polynomials of Quadratic Type Producing Strings of Primes

Abstract

AbstractThe 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.