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I think, for the question "which approach should be proposed to attack the problem", the best answer is "read Lagarias". Lagarias has a very good overview about many various attempts to the handling of the problem and gives also comments. Just a simple one, but which can at least lead to the disproof of the so-called "1-cycle","2-cycle" ... up ...


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Proving the conjecture is equivalent to finding a measure of "simplicity" of positive integers such that repeatedly applying the function in the $3x+1$ problem eventually simplifies any input. That is precisely what would be entailed in any proof, but apparently it is extremely hard to find such a thing. The $3x+1$ iteration can increase all the commonly ...



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