Acta Informatica

, Volume 43, Issue 5, pp 293–306 | Cite as

Representing the integers with powers of 2 and 3

Original Article


The Collatz sequence (n0,n1,...) is defined by nk+1 = 3nk + 1 or nk/2 depending on nk being odd or even, respectively. The Collatz conjecture (one of the most challenging open problems in Number Theory) states then that nk = 1 for some k depending on n0. This conjecture can be reformulated in a variety of ways, some of them seemingly more amenable to the methods of discrete mathematics. In this paper, we derive one such equivalent formulation involving exponential Diophantine equations. It follows that if the Collatz conjecture is true, then any number can be represented as sums of positive powers of 2 and negative powers of 3.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amigó, J.M.: Accelerated Collatz dynamics. Trabajos I+D I-2000-5, Operations Research Center of Miguel Hernandez University (available from, preprint 474) (2000)Google Scholar
  2. 2.
    Andrei S., Masalagiu C. (1998): About the Collatz conjecture. Acta Informatica 35, 167–179MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Andrei S., Kudlek M., Niculescu R.S. (2000): Some results on the Collatz problem. Acta Inform. 37(2): 145–160MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Böhm, C., Sonntacchi, G.: On the existence of cycles of a given length in integer sequences like x n+1 = x n/2 if x n even, and x n+1 = 3x n + 1 otherwise, Atti. Accad. Naz. Lincei, VIII Ser., Rend., Cl. Sci. Fis. Mat. Nat. LXIV, 260–264 (1978)Google Scholar
  5. 5.
    Chamberland, M.: An update on the 3x + 1 problem. (available from chamberl/papers.html)Google Scholar
  6. 6.
    Lagarias J.C. (1985): The 3x + 1 problem and its generalizations. Am. Math. Month. 88, 3–23MathSciNetCrossRefGoogle Scholar
  7. 7.
    Letherman S., Schleicher D., Wood R. (1999): The 3n+1-problem and holomorphic dynamics. Exp. Math. 8, 241–251MATHMathSciNetGoogle Scholar
  8. 8.
    Schroeder M.R. (1984): Number Theory in Science and Communication. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  9. 9.
    Wagon S. (1985): The Collatz Problem. The Math. Intelligencer 7, 72–76MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Centro de Investigación OperativaUniversidad Miguel HernándezElcheSpain

Personalised recommendations