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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.)
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.

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