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Collatz conjecture has been conjectured for a long time and I think there are some evidence showing that it should be true.

Similar to $P \neq NP$ conjecture, is there some interesting consequence if we assume that Collatz conjecture is true? Is there some crazy consequence if we assume that the conjecture is false?

Any reference would be appreciated.

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  • $\begingroup$ $P \neq NP$ is a much more general conjecture than the $3n+1$ (Collatz) conjecture. So refuting the latter will most likely not have the same implication-magnitude as refuting the former. As to proving - since both are assumed true, neither one of them should have too much implication anyway (expect for the "side effect" of the possibly new methods introduced or discovered during the process). $\endgroup$ – barak manos Mar 20 '15 at 22:04
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    $\begingroup$ Which parts of your question are not covered by this? $\endgroup$ – Did Mar 20 '15 at 22:34
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    $\begingroup$ Possible duplicate of What is the importance of the Collatz conjecture? $\endgroup$ – Did May 29 '16 at 15:38
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More usually with one of these intractable problems, it is not the theorem itself which has some important consequence, rather the method of proof.

The intractability of the problem represents some shortcoming in our mathematical methods, or some failure to recognise two fields or structures within mathematics to be equivalent to each other.

By the act of proving the theorem, some method is often revealed which can be applied to other, similar problems. Because of its nature, the resolution of the Collatz conjecture, if it helps in any way, will most likely extend our capacity to shortcut repetitive algebraic computations, increase our mastery of number theory and improve our understanding of how to solve some category of problems, which we generally find difficult now.

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A proof of the Collatz Conjecture is expected to provide an answer to the following question:

What is the relationship between the prime factorization of $n$ and that of $n+1$?

This would indeed provide a "missing link", as it were, between multiplicative and additive number theory.

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I suppose a solution could have serious implications on our understanding of iterated functions and dynamic systems in general. A solution to this would most likely involve some ingenious new methods of looking at dynamic systems. What the Collatz conjecture shows us is that even from simple functions we can create systems that are so chaotic mathematicians can't begin to solve it. If we could solve it, the technique would most likely be of use to solving other complex dynamical systems.

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