On the structure of the (3n+1)/2d(n) iteration problem Part I : Prediction of forward iterations

Stéphane Douady (1) , Audrey Manning (1)

1 MSC - Matière et Systèmes Complexes

To investigate the iteration of the Collatz function, we define an operation between periodic integer series that produce arithmetically subsets of them. This operation allows to decompose any periodic integer series along their generalized evenness (the number of times an integer can be divided by 2). For any periodic integer series the same regular (periodic) fractal structure is obtained. Writing how the parameters of this structure are changed through the iteration of the Collatz function, which can be simply drawn, explains the origin of the stochastic appearance of the iterations. It also allows to describe fully these iterations, and to find a general expression for them, even if still in an iterated form for the parameters. This extends the theorem of Lagarias (1985) on the periodicity of numbers of similar history. If we define the history of an integer by the successive evenness through the Collatz function iteration, and compute the number corresponding to a given history, we find that only few histories do not lead to an infinite number.

Mots clés: (3n+1)iteration; Collatz fractal; integer set structure

Domaines: Théorie des nombres [math.NT]; Systèmes dynamiques [math.DS]

Stéphane Douady, Audrey Manning. On the structure of the (3n+1)/2d(n) iteration problem Part I : Prediction of forward iterations. 2014. ⟨hal-01025833⟩