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On the finiteness of certain Rabinowitsch polynomials. (English) Zbl 0997.11024

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 R. A. Mollin [Nagoya Math. J. 105, 39-47 (1987; Zbl 0591.12005)]. Also, their Theorem 1.1 (iii) follows from the results in 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.

MSC:

11C08 Polynomials in number theory
11D85 Representation problems
11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11R09 Polynomials (irreducibility, etc.)
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References:

[1] D. Byeon, A note on class number 1 criteria for totally real algebraic number fields, Acta Arith, to appear. · Zbl 0996.11069
[2] Degert, G., Über die bestimmung der grundeinheit gewisser reellquadratischer zhalkörper, Abh. math. sem. univ. Hamburg, 22, 92-97, (1958) · Zbl 0079.05803
[3] Kutsuna, M., On a criterion for the class number of a quadratic number field to be one, Nagoya math. J., 79, 123-129, (1980) · Zbl 0447.12006
[4] Rabinowitsch, G., Eindeutigkeit der zerlegung in primzahlfaktoren in quadratischen zahlkörpern, J. reine angew. math., 142, 153-164, (1913) · JFM 44.0243.03
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