Polynomials of Quadratic Type Producing Strings of Primes
Canadian mathematical bulletin, Tome 40 (1997) no. 2, pp. 214-220
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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.
Mollin, R. A.; Goddard, B.; Coupland, S. Polynomials of Quadratic Type Producing Strings of Primes. Canadian mathematical bulletin, Tome 40 (1997) no. 2, pp. 214-220. doi: 10.4153/CMB-1997-026-5
@article{10_4153_CMB_1997_026_5,
author = {Mollin, R. A. and Goddard, B. and Coupland, S.},
title = {Polynomials of {Quadratic} {Type} {Producing} {Strings} of {Primes}},
journal = {Canadian mathematical bulletin},
pages = {214--220},
year = {1997},
volume = {40},
number = {2},
doi = {10.4153/CMB-1997-026-5},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1997-026-5/}
}
TY - JOUR AU - Mollin, R. A. AU - Goddard, B. AU - Coupland, S. TI - Polynomials of Quadratic Type Producing Strings of Primes JO - Canadian mathematical bulletin PY - 1997 SP - 214 EP - 220 VL - 40 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1997-026-5/ DO - 10.4153/CMB-1997-026-5 ID - 10_4153_CMB_1997_026_5 ER -
%0 Journal Article %A Mollin, R. A. %A Goddard, B. %A Coupland, S. %T Polynomials of Quadratic Type Producing Strings of Primes %J Canadian mathematical bulletin %D 1997 %P 214-220 %V 40 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1997-026-5/ %R 10.4153/CMB-1997-026-5 %F 10_4153_CMB_1997_026_5
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