Advertisement

Computational Methods and Function Theory

, Volume 13, Issue 2, pp 225–236 | Cite as

An Analytic Approach to the Collatz \(3n+1\) Problem for Negative Start Values

  • Lothar Berg
  • Gerhard OpferEmail author
Article
  • 225 Downloads

Abstract

In three papers, Meinardus (Über das Syracuse-Problem, Preprint Nr. 67, Universität Mannheim, 1987) and Berg and Meinardus (Results Math 25:1–12, 1994; Rostock Math Kolloq 48:11–18, 1995) have shown that the Collatz \(3n+1\) problem for positive integers \(n\) as start values can be put into the theory of complex analysis. Here we investigate the Collatz \(3n+1\) problem for negative start values. This problem is equivalent to the \(3n-1\) problem for positive start values. It is known, that this problem differs from the \(3n+1\) problem. One aspect is, that all positive start values tend, at least empirically, to either 1, 5 or 17. We describe the corresponding analytic problem for this case, where one has to show that there are not more than three linearly independent, holomorphic solutions for this problem. We conjecture that these solutions have the unit circle as natural boundary. However, the \(3n-1\) problem remains open.

Keywords

Collatz \(3n+1\) problem for negative start values \(3n-1\) problem Linear operators acting on holomorphic functions  Natural boundary 

Mathematics Subject Classification (2000)

11B37 11B83 30D05 39B32 39B72 

Notes

Acknowledgments

The research of the second mentioned author was supported by the German Science Foundation, DFG, GZ: OP 33/19-1. The same author would like to thank the librarian, Ruth Ellebracht, of the Department of Mathematics of the University of Hamburg for her permanent help in searching for the relevant literature. We thank an unknown reviewer for his (her) valuable and very detailed comments.

References

  1. 1.
    Ahlfors, L.V.: Complex Analysis. McGraw-Hill, New York (1953)zbMATHGoogle Scholar
  2. 2.
    Berg, L., Meinardus, G.: Functional equations connected with the Collatz problem. Results Math. 25, 1–12 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Berg, L., Meinardus, G.: The \(3n+1\) Collatz problem and functional equations. Rostock. Math. Kolloq. 48, 11–18 (1995)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Berg, L., Opfer, G.: An analytic approach to the Collatz \(3n+1\) problem for negative start values with an appendix of tables. Hamburger Beiträge zur Angewandten Mathematik (2012). http://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2012-11.pdf
  5. 5.
    Böhm, C., Sontacchi, G.: On the existence of cycles of 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 Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8) Mat. Appl. Vol. LXIV (1978), pp. 260–264Google Scholar
  6. 6.
    Collatz, L.: Über die Entstehung des (3n+1) Problems. J. Qufu Norm. Univ. Nat. Sci. Ed., 3, 9–11 (1986) (in Chinese). (The German title as used here was handwritten by Collatz on a copy of the Chinese paper.)Google Scholar
  7. 7.
    Crandall, R.E.: On the “\(3x+1\)” problem. Math. Comp. 32, 1281–1292 (1978)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Erdős, P.: Note on the converse of Fabry’s gap theorem. Trans. AMS 57, 102–104 (1945)Google Scholar
  9. 9.
    Fabry, E.: Sur les points singuliers d’une fonction donnée par son développement en série et l’impossibilité du prolongement analytique dans des cas très généraux. Ann. Sci. Ecole Norm. Sup. 13(sér 3), 367–399 (1896)Google Scholar
  10. 10.
    Guy, R.K.: Don’t try to solve these problems. Am. Math. Monthly 90, 35–41 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lagarias, J.C. (ed.): The Ultimate Challenge: The \(3x+1\) Problem. Amer. Math. Soc, Providence (2010)Google Scholar
  12. 12.
    Meinardus, G.: Über das Syracuse-Problem, Preprint Nr. 67, Universität Mannheim (1987)Google Scholar
  13. 13.
    Opfer, G.: An analytic approach to the Collatz \(3n+1\) problem. Hamburger Beiträge zur Angewandten Mathematik, Preprint Nr. 2011–09 (2011). http://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2011-09.pdf
  14. 14.
    Pólya, G.: On converse gap theorems. Trans. AMS 52, 65–71 (1942)zbMATHCrossRefGoogle Scholar
  15. 15.
    Wirsching, G.J.: The dynamical system generated by the \(3n+1\) function, Lecture Notes in Mathematics, vol. 1681, p. 159. Springer, Berlin (1998)Google Scholar
  16. 16.
    Wirsching, G.J.: Über das \(3n+1\) Problem. Elem. Math. 55, 142–155 (2000)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute for Mathematics, University of RostockRostockGermany
  2. 2.Faculty for Mathematics, Informatics, and Natural Sciences [MIN]University of HamburgHamburgGermany

Personalised recommendations