Polynomials of quadratic type producing strings of primes

http://dx.doi.org/10.4153/CMB-1997-026-5
Canad. Math. Bull. 40(1997), 214-220

  Published:1997-06-01
 Printed: Jun 1997

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.