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Abstract
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
defi?ne 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,
in?finitedimensional, 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 mod8^m for any
m, which amounts to arbitrarily reducing the degree of imposed coarse graining,
returns an identical conclusion.
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
defi?ne 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,
in?finitedimensional, 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 mod8^m for any
m, which amounts to arbitrarily reducing the degree of imposed coarse graining,
returns an identical conclusion.
Original language  English 

Publisher  Namur center for complex systems 
Number of pages  18 
Publication status  Submitted  21 Dec 2016 
Keywords
 number theory
 Dynamical systems
 Stochastic Processes
 Collatz conjecture
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Projects
 1 Finished

PAI n°P7/19  DYSCO: Dynamical systems, control and optimization (DYSCO)
WINKIN, J., Blondel, V., Vandewalle, J., Pintelon, R., Sepulchre, R., Vande Wouwer, A. & SARTENAER, A.
1/04/12 → 30/09/17
Project: Research
Activities

naxys seminar
Timoteo Carletti (Speaker)
25 Oct 2016Activity: Participating in or organising an event types › Participation in workshop, seminar, course

CCS 2016
Timoteo Carletti (Participant)
19 Sep 2016 → 22 Sep 2016Activity: Participating in or organising an event types › Participation in conference