Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-20T11:28:06.006Z Has data issue: false hasContentIssue false

Classification and Enumeration of Real Quadratic Fields Having Exactly One Non-Inert Prime Less Than a Minkowski Bound

Published online by Cambridge University Press:  20 November 2018

R. A. Mollin
Affiliation:
Department of Mathematics and Statistics University of Calgary Calgary, Alberta T2N 1N4 e-mail:, [email protected]
H. C. Williams
Affiliation:
Computer Science Department University of Manitoba Winnipeg, Manitoba R3T2N2 e - mail:, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We will classify those real quadratic fields K having exactly one noninert prime less than where Δ is the discriminant of K. Moreover, we will list all such K and prove that the list is complete with one possible exceptional value remaining (whose existence would be a counterexample to the Riemann hypothesis).

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Louboutin, S., Continued fractions and real quadratic fields, J. Number Theory 30(1988), 167176.Google Scholar
2. Louboutin, S., Groupes des classes d'idéaux triviaux, Acta Arith. 54(1989), 6174.Google Scholar
3. Louboutin, S., Mollin, R. A. and Williams, H. C., Class numbers of real quadratic fields, continued fractions, reduced ideals, prime-producing quadratic polynomials, and quadratic residue covers, Canadian J. Math. 44(1992), 119.Google Scholar
4. Mollin, R. A., Powers in continued fractions and class numbers of real quadratic fie Ids, Utilitas Math., to appear.Google Scholar
5. Mollin, R. A. and Williams, H. C., On prime valued polynomials and class numbers of real quadratic-fields, NagoyaMath.J. 112(1988), 143151.Google Scholar
6. Mollin, R. A. and Williams, H. C., Solution of the class number one problem for real quadratic fields of Extended Richaud-Degert type (with one possible exception). In: Number Theory, Walter de Gruyter and Co., Berlin, (1990), (ed. R. A. Mollin), 417425.Google Scholar
7. Mollin, R. A. and Williams, H. C., Class number one for real quadratic fields, continued fractions and reduced ideals. In: Number Theory and Applications, NATO ASI series C265,(ed. R. A. Mollin), (1989), 481496.Google Scholar
8. Mollin, R. A. and Williams, H. C., Computation of the Class numbers of a real quadratic field, Utilitas Math. 41(1992), 259308.Google Scholar
9. Tatuzawa, T., On a theorem of Siegel, Japan J. Math. 21(1951), 163178.Google Scholar
10. Williams, H. C. and Wunderlich, M. C., On the parallel generation of the residues for the continued fraction factoring algorithm, Math. Comp. 177(1987), 405423.Google Scholar