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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.

60G50 Sums of independent random variables; random walks
60K05 Renewal theory
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