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.
Annotation Form Interface