On the finiteness of certain Rabinowitsch polynomials
type: Item
instance of: scholarly article
title: On the finiteness of certain Rabinowitsch polynomials
zbMATH Open document ID: 0997.11024
author: Dongho Byeon
author: Harold M. Stark
published in: Journal of Number Theory
publication date: 2002-08-15T00:00:00Z
review text: The authors look at prime producing polynomials of the form \(f_m(x)= x^2+x-m\), which are called Rabinowitsch types if they are prime for a sequence of \(\lfloor \sqrt{m}\rfloor\) consecutive inputs. However, they do not mention the extensive work done in this area by the reviewer and others over the past two decades. For instance, there are dozens of references in the reviewer's book [Quadratics, CRC Press (1996; Zbl 0858.11041)]. Furthermore, their results are either already known and published or are elementary consequences of known results. For instance, their Theorems 1.1 (i)--(ii), and Theorem 1.2 are consequences of results in \textit{R. A. Mollin} [Nagoya Math. J. 105, 39-47 (1987; Zbl 0591.12005)]. Also, their Theorem 1.1 (iii) follows from the results in \textit{R. A. Mollin} [Proc. Japan Acad., Ser. A 63, 162--164 (1987; Zbl 0625.12003)]. Lastly, with respect to their final theorem, they end the paper by stating: ``For the present, it seems difficult to extend Theorem 1.3 to arbitrary \(1+4m\)''. However, in a letter sent to them by Stephane Louboutin, it is pointed out that the general result is easily proved in a few lines using the most elementary techniques.
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11D85
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R09
zbMATH DE Number: 1781939
zbMATH Keywords: class number
zbMATH Keywords: quadratic field
zbMATH Keywords: prime producing polynomials
reviewed by: Richard A. Mollin
MaRDI profile type: Publication
full work available at URL: jnth.2001.2729
OpenAlex ID: W4213063501
cites work: A note on class number 1 criteria for totally real algebraic number fields
cites work: Über die Bestimmung der Grundeinheit gewisser reell-quadratischer Zahlkörper
cites work: On a criterion for the class number of a quadratic number field to be one
DOI: 10.1006/JNTH.2001.2729
description: scientific article; zbMATH DE number 1781939