Quantifying the degree of average contraction of Collatz orbits

Timoteo Carletti, Duccio Fanelli

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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.
Original languageEnglish
Pages (from-to)445-468
Number of pages24
JournalBollettino dell'Unione Matematica Italiana
Issue number4
Publication statusPublished - 3 Oct 2017


  • Collatz conjecture
  • Number theory
  • Markov process
  • Ergodic dynamical systems


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  • CCS 2016

    Timoteo Carletti (Speaker)

    22 Sep 2016

    Activity: Participating in or organising an event typesParticipation in conference

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