Continued Fractions of Algebraic Numbers

  • Enrico Bombieri
  • Alfred J. van der Poorten
Part of the Mathematics and Its Applications book series (MAIA, volume 325)


The work of the second author was supported in part by grants from the Australian Research Council and by a research agreement with Digital Equipment Corporation.


Rational Approximation Continue Fraction Algebraic Number Digital Equipment Corporation Continue Fraction Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. D. Bryuno, Continued fraction expansion of algebraic numbers, Zh. Vychisl. Mat. i. Mat. Fiz. 4 (1964), 211–221. English translation: USSR Comput. Math. and Math. Phys. 4 (1964), 1–15.Google Scholar
  2. [2]
    David G. Cantor, Paul H. Galyean and Horst Günter Zimmer, A continued fraction algorithm for real algebraic numbers, Math. Comp. 26 (1972), 785–791.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    E. A. Dzenskevich and A. P. Shapiro, On the expansion of irrationalities of third and fourth degree into a continued fraction, in: Algorithmic and numerical problems in algebra and number theory (Russian), 3–9, 87, Akad. Nauk. SSSR, Dal’nevostochn. Otdel., Vladivostok, 1987.Google Scholar
  4. [4]
    Donald E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Menlo-Park: Addison—Wesley, 1981.Google Scholar
  5. [5]
    Serge Lang and Hale Trotter, Continued fractions for some algebraic numbers,J. für Math. 255 (1972), 112–134; Addendum: ibid. 219–220.Google Scholar
  6. [6]
    L. Lewin, Dilogarithms and associated functions, London: Macdonald, (1958).zbMATHGoogle Scholar
  7. [7]
    J. von Neumann and B. Tuckerman, Continued fraction expansion of 21, Math. Tables Aids Comput. 9 (1955), 23–24.MathSciNetGoogle Scholar
  8. [8]
    A. J. van der Poorten, A proof that Euler missed... Apéry’s proof of the irrationality of ((3); An informal report, Math. Intelligencer 1 (1979), 195–203.Google Scholar
  9. [9]
    A. J. van der Poorten, Some wonderful formulas... an introduction to polylogarithms, Queen’s Papers in Pure and Applied Mathematics 54 (1980), 269–286.Google Scholar
  10. [10]
    R. D. Richtmyer, M. Devaney and N. Metropolis, Continued fraction expansions of algebraic numbers, Numer. Math. 4 (1962), 68–84.MathSciNetzbMATHGoogle Scholar
  11. [11]
    K. F. Roth, Rational approximations to algebraic numbers,Mathematika 2 (1955), 1–20; Corrigendum: ibid. 168.Google Scholar
  12. [12]
    H. M. Stark, An explanation of some exotic continued fractions found by Brillhart, in: A. O. L. Atkin and B. J. Birch (eds.), Computers in Number Theory, London: Academic Press, 1971, 21–35.Google Scholar
  13. [13]
    J. V. Uspensky, Theory of Equations, McGraw—Hill Book Company, 1948.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Enrico Bombieri
    • 1
  • Alfred J. van der Poorten
    • 2
  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.CeNTRe for Number Theory ResearchMacquarie UniversityAustralia

Personalised recommendations