This article is cited in 10 papers

Short Communications

Estimates for the Syracuse problem via a probabilistic model

K. A. Borovkov^a, D. Pfeifer<sup>b

a</sup> University of Melbourne, Department of Mathematics and Statistics
^b Institut für Mathematische Stochastik, Universität, Germany

Abstract: We employ a simple stochastic model for the Syracuse problem (also known as the $(3x+ 1)$ problem) to get estimates for the average behavior of the trajectories of the original deterministic dynamical system. The use of the model is supported not only by certain similarities between the governing rules in the systems, but also by a qualitative estimate of the rate of approximation. From the model, we derive explicit formulae for the asymptotic densities of some sets of interest for the original sequence. We also approximate the asymptotic distributions for the stopping times (times until absorption in the only known cycle $\{1,2\}$) of the original system and give numerical illustrations of our results.

Keywords: Syracuse problem, dynamical system, random walk.

Language: English

DOI: 10.4213/tvp472

 English version: Theory of Probability and its Applications, 2001, 45:2, 300–310

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