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The following heuristic probabilistic argument supports the Conjecture (see ). 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.)

From the viewpoint of this heuristic argument, the central difficulty of the problem lies in understanding in detail the ``mixing'' properties of iterates of the function for all powers of 2. The function does indeed have some ``mixing'' properties given by Theorems B and K below; these are much weaker than what one needs to settle the Conjecture.  Contents Next: Behavior of the Up: The 3x+1 problem. Previous: The 3x+1 problem.