Abstract
The oddness of the length of the period of the continued fraction expansion of the square root of an odd prime integer equal to 3 modulo 4 is well known. We determine its value modulo 4. We also give a similar result for the square root of twice an odd prime integer equal to 3 modulo 4.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
D. Chakraborty, A. Saikia, Explicit solutions to \(u^2-pv^2=\pm 2\) and a conjecture of Mordell, 14 October 2018, arXiv:1810.05980v1
E.P. Golubeva, Quadratic irrationals with fixed period length in the continued fraction expansion. J. Math. Sci. 70, 2059–2076 (1994)
H. Hasse, Vorlesungen über Zahlentheorie. Zweite neubearbeitete Auflage. Die Grundlehren der Mathematischen Wissen schaften, Band, vol. 59 (Springer, Berlin, 1964)
F. Kawamoto, Y. Kishi, K. Tomita, Continued fraction expansions with even period and primary symmetric parts with extremely large end. Comment. Math. Univ. St. Pauli 64(2), 131–155 (2015)
S. Louboutin, Continued fractions and real quadratic fields. J. Number Theory 30, 167–176 (1988)
S. Louboutin, Groupes des classes d’idéaux triviaux. Acta Arith. 54, 61–74 (1989)
Perron, Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche. 3te Aufl (B. G. Teubner Verlagsgesellschaft, Stuttgart, 1954)
A. Saikia, On the continued fraction of \(\sqrt{p}\), in International Conference on Class Groups of Number Fields and Related Topics, Harish-Chandra Research Institute, Allahabad, Inde, 9 October 2018
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Louboutin, S.R. (2020). On the Continued Fraction Expansions of \(\sqrt{p}\) and \(\sqrt{2p}\) for Primes \(p\equiv 3\pmod 4\). In: Chakraborty, K., Hoque, A., Pandey, P. (eds) Class Groups of Number Fields and Related Topics. Springer, Singapore. https://doi.org/10.1007/978-981-15-1514-9_16
Download citation
DOI: https://doi.org/10.1007/978-981-15-1514-9_16
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-1513-2
Online ISBN: 978-981-15-1514-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)