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.
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