Monatshefte für Mathematik

, Volume 101, Issue 4, pp 309–315

Dyadic fractions with small partial quotients

  • Harald Niederreiter
Article

Abstract

It is proved that ifm is a power of 2, then there exists an odd integera with 1≤a<m such that all partial quotients in the continued fraction expansion ofa/m are bounded by 3. The upper bound 3 is best possible. Similar results can be shown for powers of other small numbers.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Borosh, I., Niederreiter, H.: Optimal multipliers for pseudo-random number generation by the linear congruential method. BIT23, 65–74 (1983).Google Scholar
  2. [2]
    Cusick, T. W.: Continuants with bounded digits. Mathematika24, 166–172 (1977).Google Scholar
  3. [3]
    Cusick, T. W.: Continuants with bounded digits—II. Mathematika25, 107–109 (1978).Google Scholar
  4. [4]
    Cusick, T. W.: Continuants with bounded digits—III. Mh. Math.99, 105–109 (1985).Google Scholar
  5. [5]
    Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers. 4th ed. Oxford: Clarendon Press. 1960.Google Scholar
  6. [6]
    Hlawka, E.: Zur angenäherten Berechnung mehrfacher Integrale. Mh. Math.66, 140–151 (1962).Google Scholar
  7. [7]
    Niederreiter, H.: Pseudo-random numbers and optimal coefficients. Adv. Math.26, 99–181 (1977).Google Scholar
  8. [8]
    Niederreiter, H.: Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Amer. Math. Soc.84, 957–1041 (1978).Google Scholar
  9. [9]
    Pethö, A.: Perfect powers in second order linear recurrences. J. Number Theory15, 5–13 (1982).Google Scholar
  10. [10]
    Ramharter, G.: Some metrical properties of continued fractions. Mathematika30, 117–132 (1983).Google Scholar
  11. [11]
    Zaremba, S. K.: Good lattice points, discrepancy, and numerical integration. Ann. Mat. Pura Appl. (IV)73, 293–317 (1966).Google Scholar
  12. [12]
    Zaremba, S. K.: La méthode des “bons treillis” pour le calcul des intégrales multiples. In: Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), pp. 39–119. New York: Academic Press. 1972.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Harald Niederreiter
    • 1
  1. 1.Kommission für MathematikÖsterreichische Akademie der WissenschaftenViennaAustria

Personalised recommendations