zbMATH — the first resource for mathematics

Estimates for the Syracuse problem via a probabilistic model. (English) Zbl 0984.60050
Theory Probab. Appl. 45, No. 2, 300-310 (2000) and Teor. Veroyatn. Primen. 45, No. 2, 386-395 (2000).
The Syracuse problem, or the Collatz’ problem, Kakutani’s problem, Ulam’s problem, Hasse’s algorithm, also called the $$3x+1$$ problem, is investigated via a probabilistic model. Roughly speaking the problem concerns the behaviour of a dynamical system whose orbit is generated by the function $$f(x)= {3x+1\over 2}$$ if $$x$$ is odd and $$f(x)= {x\over 2}$$ if $$x$$ is even. The problem amounts to verifying the conjecture that $$t(x_0)= \inf\{k\geq 1: x_k= 1\}$$ is finite, where $$x_0$$ is the starting point and $$\{x_n\}$$ the orbit. The paper investigates certain density properties of the orbit which are related to the conjecture.

MSC:
 60G50 Sums of independent random variables; random walks 60K05 Renewal theory
Full Text: