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Please explain to me, what do Terence Tao and other mathematicians who write books about the Collatz Conjecture, what do they want to do in general? Do they want to prove this hypothesis, or disprove it?

What can we generally conclude from these books? what are the mathematical predictions about this problem ,what is it in general?

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  • $\begingroup$ They contain what we currently know about Collatz conjecture. For example, Steiner (1977) proved that there is no 1-cycle other than the trivial (1;2); Simons & de Weger (2003) extended this proof up to 68-cycles: there is no k-cycle up to k = 68. Knowledge of this sort makes finding counter-examples easier. $\endgroup$
    – Kenny Lau
    Sep 9, 2017 at 14:33
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    $\begingroup$ I don't know that there is a clear consensus on the Collatz conjecture. There are some heuristics which make it plausible, but I don't think anyone would be too terribly shocked by some big counterexample. At the moment, all we can do is to search for counterexamples, try to prove what we can, and make predictions based on it which we can then test. $\endgroup$
    – lulu
    Sep 9, 2017 at 14:38
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    $\begingroup$ @Student I agree lulu , I also tend to believe that it is true. But to be honest, I would find a counterexample (or just a disprove) more interesting although a proof would be truely sensational. $\endgroup$
    – Peter
    Sep 9, 2017 at 15:55
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    $\begingroup$ Although they may not have solved the problem in question, they generally want to share their learnings and insights, what they conjecture might be approaches to tackle the problem, and perhaps some of the reasons why they believe the problem is difficult. $\endgroup$ Sep 10, 2017 at 18:11
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    $\begingroup$ @lulu I think most who have studied the problem in any depth would be surprised by any counterexample. The reason is... there can not be some isolated counterexample. The smallest counterexample would be the base of a tree which branches at the same rate as the tree known to branch from 1 and its predecessors must occupy the ranges of numbers occupied by the trivial tree. Furthermore, if it did not terminate in a loop it would also have infinitely many successors which would again have to occupy the same ranges of numbers, every odd successor also having infinitely many other predecessors $\endgroup$ Sep 10, 2017 at 18:22

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There is The Ultimate Challenge: the 3x+1 Problem by Jeffrey Lagarias (which is a collection of papers by various writers). There are papers and chapters in books by others.

But generally, the books on the Collatz conjecture seem to be the self-published writings of cranks.

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  • $\begingroup$ What is crank?? Ohh sorry English is not my native language.. $\endgroup$
    – Student
    Sep 9, 2017 at 14:48
  • $\begingroup$ An interesting look at cranks is here: amazon.com/Mathematical-Cranks-Spectrum-Underwood-Dudley/dp/… $\endgroup$
    – GEdgar
    Sep 9, 2017 at 14:51
  • $\begingroup$ Finally...,what is your personal thought about this conjecture? can I ask?? $\endgroup$
    – Student
    Sep 9, 2017 at 15:18

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