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.
On
In this nice presentation from Terry Tao https://terrytao.files.wordpress.com/2020/02/collatz.pdf (a video of him presenting these slides can be found here: https://www.youtube.com/watch?v=X2p5eMWyaFs), he mentions that "the absence of non-trivial Collatz cycles can be shown to imply a difficult result in number theory", namely the fact that "the gap between powers of 2 and powers of 3 goes to infinity."
Tao also writes "This theorem is known to be true, but its proof is difficult, requiring a deep result known as Baker’s theorem (which earned Alan Baker the Fields medal in 1970). So solving the Collatz conjecture may be at least as hard as proving Baker’s theorem!"
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.