Journal of Soviet Mathematics

, Volume 18, Issue 6, pp 919–923

Length of the period of a quadratic irrational

  • E. V. Podsypanin
Article

Abstract

Let ξ be a real quadratic irrational of discriminant D=f2Di>0, where Di is the fundamental discriminant of the field
and h are the character and the number of classes of the field
, respectively, and\(L\left( {1,\chi } \right) = \sum\limits_{n = 1}^\infty {\frac{{\chi \left( n \right)}}{n}} \) proves the following estimate for the length l of the period of the expansion of ξ into a continued fraction:
where ω=1 if f=1 and ω=2 if f>1. A. S. Pen and B. F. Skubenko (Mat. Zametki, 5, No. 4, 413–482 (1969)) have proved this estimate in the case f=1, D1≡0 (mod4).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature cited

  1. 1.
    Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York (1966).Google Scholar
  2. 2.
    L. V. Danilov and G. V. Danilov, “An estimate from above for the period of quadratic irrational,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat.,19, No. 9, 19–24 (1975).Google Scholar
  3. 3.
    E. A. Murzaev, “On an estimate of the upper bound of the length of the period of a continued fraction,” Volzhskii Mat. Sb., No. 5, 260–261 (1966).Google Scholar
  4. 4.
    A. S. Pen and B. F. Skubenko, “An upper bound for the period of a quadratic irrationality,” Mat. Zametki,5, No. 4, 413–418 (1969).Google Scholar
  5. 5.
    E. V. Podsypanin, “An estimate from above for the length of the period of a quadratic irrationality,” in: Tezisy Dok. i Soobshch. Vsesoyuz. Shk. po Teorii Chisel, Dushanbe (1977), p. 100.Google Scholar
  6. 6.
    J. H. E. Cohn, “The length of the period of the simple continued fraction of\(\sqrt d \),” Pac. J. Math.,71, No. 1, 21–32 (1977).Google Scholar
  7. 7.
    D. R. Hickerson, “Length of period of simple continued fraction expansion of,” Pac. J. Math.,46, 429–432 (1973).Google Scholar
  8. 8.
    K. E. Hirst, “The length of periodic continued fractions,” Monatsh. Math.,76, 428–435 (1972).Google Scholar
  9. 9.
    R. C. Stanton, C. Sudler, Jr., and H. C. Williams, “An upper bound for the period of the simple continued fraction for,” Pac. J. Math.,67, No. 2, 525–536 (1976).Google Scholar

Copyright information

© Plenum Publishing Corporation 1982

Authors and Affiliations

  • E. V. Podsypanin

There are no affiliations available

Personalised recommendations