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* Is the ***3x+1** problem intractably hard?
The difficulty of settling the **3x+1** problem seems connected to the
fact that it is a deterministic process that simulates ``random''
behavior.
We face this dilemma:
On the one hand, to the extent that the problem has structure,
we can analyze it ---
yet it is precisely this structure that seems to prevent us from proving
that it behaves ``randomly.''
On the other hand, to the extent that the problem is structureless
and ``random,''
we have nothing to analyze and consequently cannot rigorously prove anything.
Of course there remains the possibility that someone will find some hidden
regularity in the **3x+1** problem that allows some of the conjectures about it to be settled.
The existing general methods in number theory and ergodic theory do not
seem to touch the **3x+1** problem; in this sense it
seems intractable at present.
Indeed all the conjectures made in this paper seem currently to be out of
reach if they are true; I think there is more chance of
disproving those that are false.
* If the ***3x+1** problem is intractable, why should one bother to study it?
One answer is provided by the following aphorism: ``No problem is so
intractable that something interesting cannot be said about it.''
Study of the **3x+1** problem has uncovered a number of interesting phenomena;
I believe further study of it may be rewarded by the discovery of
other new phenomena.
It also serves as a benchmark to measure the progress of general
mathematical theories.
For example,
future developments in solving exponential Diophantine equations may lead to the
resolution of the Finite Cycles Conjecture.

* If all the conjectures made in this paper are intractable, where would one
begin to do research on this deceptively simple problem?*
As a guide to doing research, I ask questions.
Here are a few that occur to me:
For the **3x+1** problem, what restrictions are there on the growth in size
of members of a divergent trajectory assuming that one exists?
What interesting properties does the function have?
Is there some direct characterization of
other than the recursive definition (2.33)?

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