Quantifying the degree of average contraction of Collatz orbits



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 


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© 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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