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The known results on the problem are most elegantly expressed in terms of iterations of the function One way to think of the problem involves a directed graph whose vertices are the positive integers and that has directed edges from

We call the sequence of iterates
the * trajectory* of **n**.
There are three possible behaviors for such trajectories when **n > 0**.

- (i).
*Convergent trajectory.*Some . - (ii).
*Non-trivial cyclic trajectory.*The sequence eventually becomes periodic and

for any . - (iii).
*Divergent trajectory.*.

The appeal of the problem lies in the irregular behavior of the
successive iterates .
One can measure this behavior using the stopping time, the total stopping
time, and the
* expansion factor*
defined by

The Conjecture has been numerically checked for a large range of
values of **n**.
It is an interesting problem to find efficient algorithms to test the
conjecture on a computer.
The current record for verifying the Conjecture seems to be held
by Nabuo Yoneda at the University of Tokyo,
who has reportedly checked it for all
[2].
In several places the statement appears that A. S. Fraenkel has checked that
all have a finite total stopping time;
this statement is erroneous [32].

- A heuristic argument.
- Behavior of the stopping time function.
- What is the relation between the coefficient stopping time and the stopping time?
- How many elements don't have a finite stopping time?
- Behavior of the total stopping time function.
- Are there non-trivial cycles?
- Do divergent trajectories exist?
- Connections of the problem to ergodic theory.

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