Bollettino dell'Unione Matematica Italiana

, Volume 11, Issue 4, pp 445–468 | Cite as

Quantifying the degree of average contraction of Collatz orbits

  • Timoteo Carletti
  • Duccio Fanelli


We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the \((3x+1)\) or Syracuse conjecture. The analysis is structured as follows. First, three distinct fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which define the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8 (mod8), obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates in the original, infinite-dimensional, space of positive integers. Then, we consider a Markov chain which runs on the reduced space of mod8 congruence classes of integers. The transition probabilities of the Markov chain are computed from the deterministic map, by employing a measure that is invariant for the map itself. Working in this setting, we demonstrate that the stationary distribution sampled by the stochastic system induces a contracting behaviour for the orbits of the deterministic map on the original space of the positive integers. Sampling the equilibrium distribution on the congruence classes mod \(8^m\) for any m, which amounts to arbitrarily reducing the degree of imposed coarse graining, returns an identical conclusion.


Collatz conjecture Number theory Markov process Ergodic dynamical systems 



We would like to thank the numerous colleagues who interacted with us all along the various stages of the writing of this work. In particular we warmly thank Claudio Bonanno, Carlo Carminati, Jean-Charle Delvenne, Craig Alan Feinstein, Steffen Kionke, Shlomo Levental, Stefano Marmi, Vassilis Papanicolaou, François Stealens and Cédric Villani, for their insightful comments and remarks. The work of T.C. presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office.


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Copyright information

© Unione Matematica Italiana 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Namur Institute for Complex Systems-naXysUniversity of NamurNamurBelgium
  2. 2.Dipartimento di Fisica e AstronomiaUniversity of Florence, INFN and CSDCFlorenceItaly

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