Abstract
We investigate the class number one problem for a parametric family of real quadratic fields of the form \(\mathbb {Q}( \sqrt{m^2+4r})\) for certain positive integers m and r.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ankeny, N.C., Chowla, S., Hasse, H.: On the class-number of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math. 217, 217–220 (1965)
Biró, A.: Yokoi’s conjecture. Acta Arith. 106, 85–104 (2003)
Biró, A., Lapkova, K.: The class number one problem for the real quadratic fields \(\mathbb{Q}(\sqrt{(an)^2+4a})\). Acta Arith. 172(2), 117–131 (2016)
Byeon, D., Kim, H.K.: Class number \(1\) criteria for real quadratic fields of Richaud–Degert type. J. Number Theory 57, 328–339 (1996)
Chakraborty, K., Hoque, A., Mishra, M.: A note on certain real quadratic fields with class number up to three. Kyushu J. Math. 74(1), 201–210 (2020)
Louboutin, S.: Continued fractions and real quadratic fields. J. Number Theory 30(2), 167–176 (1988)
Louboutin, S.: Prime producing quadratic polynomials and class-numbers of real quadratic fields. Can. J. Math. 42(2), 315–341 (1990)
Mollin, R.A., 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 (Banff, AB, 1988), pp. 417–425. de Gruyter, Berlin (1990)
Yokoi, H.: Class number one problem for certain kind of real quadratic fields. In: Proceedings of the International Conference on Class Numbers and Fundamental Units of Algebraic Number Fields (Katata, 1986), pp. 125–137. Nagoya University, Nagoya (1986)
Yokoi, H.: The fundamental unit and class number one problem of real quadratic fields with prime discriminant. Nagoya Math. J. 120, 51–59 (1990)
Acknowledgements
The authors are grateful to the anonymous referee for careful reading, pointing out a serious error in the previous version, and valuable comments which have helped to improve this paper. The authors are also grateful to the referee for drawing the papers [1, 6] to their attention. The authors acknowledge the grants SERB-NPDF (PDF/2017/001958) and SERB MATRICS Project No. MTR/2017/00100, Govt. of India.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hoque, A., Kotyada, S. Class number one problem for the real quadratic fields \({{\mathbb {Q}({\sqrt{m^2+2r}})}}\). Arch. Math. 116, 33–36 (2021). https://doi.org/10.1007/s00013-020-01520-w
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-020-01520-w