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


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.


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 



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.


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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

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