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''
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
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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