Estimates for the Syracuse problem via a probabilistic model (Q2752956)

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.