×

zbMATH — the first resource for mathematics

The Rabinowitsch-Mollin-Williams theorem revisited. (English) Zbl 1290.11151
Summary: We completely classify all polynomials of type \((x^2+x - (\Delta - 1))/4\) which are prime or 1 for a range of consecutive integers \(x\geq 0\), called Rabinowitsch polynomials, where \(\Delta \equiv 1\pmod 4\) with \(\Delta >1\) square-free. This corrects, extends, and completes the results by Byeon and Stark via the use of an updated version of what Andrew Granville has dubbed the Rabinowitsch-Mollin-Williams Theorem by A. Granville and the author [Acta Arith. 96, No. 2, 139–153 (2000; Zbl 0985.11040) and the author[Quadratics. Boca Raton, FL: CRC Press (1996; Zbl 0858.11001). Furthermore, we verify conjectures of this author and pose more based on the new data.

MSC:
11R29 Class numbers, class groups, discriminants
11R09 Polynomials (irreducibility, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] G. Rabinowitsch, “Eindeutigkeit der zerlegung in primzahlfaktoren in quadratischen Zahlkörpern,” Journal für die Reine und Angewandte Mathematik, vol. 142, pp. 153-164, 1913. · JFM 44.0243.03
[2] R. A. Mollin and H. C. Williams, “On prime valued polynomials and class numbers of real quadratic fields,” Nagoya Mathematical Journal, vol. 112, pp. 143-151, 1988. · Zbl 0629.12004
[3] R. A. Mollin and H. C. Williams, “Prime producing quadratic polynomials and real quadratic fields of class number one,” in Théorie des Nombres (Quebec, PQ, 1987), pp. 654-663, de Gruyter, Berlin, Germany, 1989. · Zbl 0695.12002
[4] R. A. Mollin, Quadratics, CRC Press Series on Discrete Mathematics and Its Applications, CRC Press, Boca Raton, Fla, USA, 1996. · Zbl 0858.11001
[5] R. A. Mollin, “An elementary proof of the Rabinowitsch-Mollin-Williams criterion for real quadratic fields,” Journal of Mathematical Sciences, vol. 7, no. 1, pp. 17-27, 1996. · Zbl 1286.11171
[6] D. Byeon and H. M. Stark, “On the finiteness of certain Rabinowitsch polynomials,” Journal of Number Theory, vol. 94, no. 1, pp. 219-221, 2002. · Zbl 1033.11010
[7] D. Byeon and H. M. Stark, “On the finiteness of certain Rabinowitsch polynomials. II,” Journal of Number Theory, vol. 99, no. 1, pp. 177-180, 2003. · Zbl 1033.11010
[8] R. A. Mollin, “Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S. Chowla,” Proceedings of the American Mathematical Society, vol. 102, no. 1, pp. 17-21, 1988. · Zbl 0673.12005
[9] D. Byeon, M. Kim, and J. Lee, “Mollin’s conjecture,” Acta Arithmetica, vol. 126, no. 2, pp. 99-114, 2007. · Zbl 1125.11059
[10] R. A. Mollin, Fundamental Number Theory with Applications, Discrete Mathematics and Its Applications, Chapman & Hall/CRC, Taylor and Francis, Boca Raton, Fla, USA, 2nd edition, 2008. · Zbl 1175.11001
[11] R. A. Mollin, Algebraic Number Theory, Discrete Mathematics and Its Applications, Chapman & Hall/CRC, Taylor and Francis, Boca Raton, Fla, USA, 1999. · Zbl 0930.11001
[12] R. A. Mollin, Fundamental Number Theory with Applications, Chapman & Hall/CRC, Taylor and Francis, Boca Raton, Fla, USA, 1st edition, 1998. · Zbl 1175.11001
[13] S. Louboutin, R. A. Mollin, and H. C. Williams, “Class numbers of real quadratic fields, continued fractions, reduced ideals, prime-producing quadratic polynomials and quadratic residue covers,” Canadian Journal of Mathematics, vol. 44, no. 4, pp. 824-842, 1992. · Zbl 0771.11039
[14] A. Biró, “Chowla’s conjecture,” Acta Arithmetica, vol. 107, no. 2, pp. 179-194, 2003. · Zbl 1154.11339
[15] P. Ribenboim, The Book of Prime Number Records, Springer, New York, NY, USA, 1988. · Zbl 0642.10001
[16] A. Biró, “Yokoi’s conjecture,” Acta Arithmetica, vol. 106, no. 1, pp. 85-104, 2003. · Zbl 1154.11338
[17] J.-C. Schlage-Puchta, “Finiteness of a class of Rabinowitsch polynomials,” Archivum Mathematicum, vol. 40, no. 3, pp. 259-261, 2004. · Zbl 1122.11070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.