Monatshefte für Mathematik

, Volume 111, Issue 2, pp 119–126 | Cite as

Continued fractions for some alternating series

  • J. L. Davison
  • J. O. Shallit


We discuss certain simple continued fractions that exhibit a type of “self-similar” structure: their partial quotients are formed by perturbing and shifting the denominators of their convergents. We prove that all such continued fractions represent transcendental numbers. As an application, we prove that Cahen's constant
$$C = \sum\limits_{i \geqslant 0} {\frac{{( - 1)^i }}{{S_i - 1}}}$$
is transcendental. Here (S n ) isSylvester's sequence defined byS0=2 andS n+1 =S n 2 S n +1 forn≥0. We also explicitly compute the continued fraction for the numberC; its partial quotients grow doubly exponentially and they are all squares.


Continue Fraction Partial Quotient Transcendental Number 
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]
    Adams, W. W., Davison, J. L.: A remarkable class of continued fractions. Proc. Amer. Math. Soc.65, 194–198 (1977).Google Scholar
  2. [2]
    Aho, A. V., Sloane, N. J. A.: Some doubly exponential sequences. Fib. Quart.11, 429–437 (1973).Google Scholar
  3. [3]
    Blanchard, A., Mendès France, M.: Symétrie et transcendance. Bull. Sc. Math.106, 325–335 (1982).Google Scholar
  4. [4]
    Böhmer, P. E.: Über die Transzendenz gewisser dyadischer Brüche. Math. Ann.96, 367–377 (1926).Google Scholar
  5. [5]
    Cahen, E.: Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues. Nouvelles Annales de Mathématiques10, 508–514 (1891).Google Scholar
  6. [6]
    Carmichael, R. D.: Diophantine Analysis. New York: John Wiley & Sons. 1915.Google Scholar
  7. [7]
    Curtiss, D. R.: On Kellogg's diophantine problem. Amer. Math. Monthly29, 380–387 (1922).Google Scholar
  8. [8]
    Curtiss, D. R.: Classes of diophantine equations whose positive integral solutions are bounded. Bull. Amer. Math. Soc.35, 859–865 (1929).Google Scholar
  9. [9]
    Danilov, L. V.: Some classes of transcendental numbers. Matematicheskie Zametki12, 149–154 (1972). (English translation in Math. Notes Acad. Sci. USSR12, 524–527 (1972)).Google Scholar
  10. [10]
    Davison, J. L.: A series and its associated continued fraction. Proc. Amer. Math. Soc.63, 29–32 (1977).Google Scholar
  11. [11]
    Erdös, P.: Az\(\frac{1}{{x_1 }} + \frac{1}{{x_2 }} + \cdots \frac{1}{{x_n }} = \frac{a}{b}\) egyenlet egész számú megoldásairól. Mat. Lapok1, 192–210 (1950).Google Scholar
  12. [12]
    Erdös, P., Straus, E. G.: On the irrationality of certain Ahmes series. J. Indian Math. Soc.27, 129–133 (1964).Google Scholar
  13. [13]
    Franklin, J. N., Golomb, S. W.: A function-theoretic approach to the study of nonlinear recurring sequences. Pacific J. Math.56, 455–468 (1975).Google Scholar
  14. [14]
    Golomb, S. W.: On the sum of the reciprocals of the Fermat numbers and related irrationalities. Canad. J. Math.15, 475–478 (1963).Google Scholar
  15. [15]
    Golomb, S. W.: On certain nonlinear recurring sequences. Amer. Math. Monthly70, 403–405 (1963).Google Scholar
  16. [16]
    Greene, D. H., Knuth, D. E.: Mathematics for the Analysis of Algorithms. Basel: Birkhäuser. 1982.Google Scholar
  17. [17]
    Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers. Oxford. Univ. Press. 1985.Google Scholar
  18. [18]
    Hensley, D.: Lattice vertex polytopes with few interior lattice points. Pacific J. Math.105, 183–191 (1983).Google Scholar
  19. [19]
    Kellogg, O. D.: On a diophantine problem. Amer. Math. Monthly28, 300–303 (1921).Google Scholar
  20. [20]
    Kmošek, M.: Rozwini ecie niektórych liczb niewymiernych na ułamki łańcuchowe. (Master's Thesis). Warsaw: Uniwersytet Warszawski. 1979.Google Scholar
  21. [21]
    Köhler, G.: Some more predictable continued fractions. Mh. Math.89, 95–100 (1980).Google Scholar
  22. [22]
    Lagarias, J. C., Ziegler, G. M.: Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Canad. J. Math., to appear.Google Scholar
  23. [23]
    Odoni, R. W. K.: On the prime divisors of the sequencew n+1=1+w 1...w n. J. London Math. Soc.32, 1–11 (1985).Google Scholar
  24. [24]
    Pethö, A.: Simple continued fractions for the Fredholm numbers. J. Number Theory14, 232–236 (1982).Google Scholar
  25. [25]
    van der Poorten, A. J., Shallit, J. O.: Folded continued fractions. Preprint. 1990.Google Scholar
  26. [26]
    Remez, E. Ya.: On series with alternating signs which may be connected with two algorithms of M. V. Ostrogradskiî for the approximation of irrational numbers. Uspekhi Mat. Nauk6 (No. 5), 33–42 (1951).Google Scholar
  27. [27]
    Roberts, J.: Elementary Number Theory. MIT Press. 1977.Google Scholar
  28. [28]
    Roth, K. F.: Rational approximations to algebraic numbers. Mathematika2, 1–20 (1955).Google Scholar
  29. [29]
    Salzer, H. E.: The approximation of numbers as sums of reciprocals. Amer. Math. Monthly54, 135–142 (1947).Google Scholar
  30. [30]
    Salzer, H. E.: Further remarks on the approximation of numbers as sums of reciprocals. Amer. Math. Monthly53, 350–356 (1948).Google Scholar
  31. [31]
    Shallit, J. O.: Simple continued fractions for some irrational numbers. J. Number Theory11, 209–217 (1979).Google Scholar
  32. [32]
    Shallit, J. O.: Simple continued fractions for some irrational numbers II. J. Number Theory14, 228–231 (1982).Google Scholar
  33. [33]
    Shallit, J. O.: Sylvester's sequence and the transcendence of Cahen's constant. In: The Mathematical Heritage of Carl Friedrich Gauss. World Scientific Publishing. To appear.Google Scholar
  34. [34]
    Sloane, N. J. A.: A Handbook of Integer Sequences. New York: Academic Press. 1973.Google Scholar
  35. [35]
    Sylvester, J. J.: On a point in the theory of vulgar fractions. Amer. J. Math.3, 332–334 (1880).Google Scholar
  36. [36]
    Sylvester, J. J.: Postscript to note on a point in vulgar fractions. Amer. J. Math.3, 388–389 (1880).Google Scholar
  37. [37]
    Takenouchi, T.: On an indeterminate equation. Proc. Physico-Mathematical Soc. Japan3, 78–92 (1921).Google Scholar
  38. [38]
    Tamura, J.: Symmetric continued fractions related to certain series. Manuscript. 1990.Google Scholar
  39. [39]
    Zaks, J., Perles, M. A., Wills, J. M.: On lattice polytopes having interior lattice points. Elem. Math.37, 44–46 (1982).Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • J. L. Davison
    • 1
  • J. O. Shallit
    • 2
  1. 1.Mathematics and Computer ScienceLaurentian UniversitySudburyCanada
  2. 2.Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations