The following heuristic probabilistic argument supports the Conjecture (see [28]). Pick an odd integer at random and iterate the function T until another odd integer occurs. Then of the time of the time of the time , and so on. If one supposes that the function T is sufficiently ``mixing'' that successive odd integers in the trajectory of n behave as though they were drawn at random from the set of odd integers for all k, then the expected growth in size between two consecutive odd integers in such a trajectory is the multiplicative factor

Consequently this heuristic argument suggests that on average the iterates in a trajectory tend to shrink in size, so that divergent trajectories should not exist. Furthermore it suggests that the total stopping time is (in some average sense) a constant multiple of . (Click here for more.)
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