Where do mathematicians fail while trying to prove the Collatz conjecture?



I am by no means a mathematician, and I understand seemingly small changes can make huge differences like the XKCD of the difference between checking if a picture was taken in a park vs if a picture is of a bird.

If the proof is required to show that all numbers will eventually reach a 2^n value which will bring you to a value of one, it seems like that should be possible by proving you can reach every value from a 2^n value. While I don’t have the education to figure that out, it doesn’t seem to be as difficult as some of the crazy proofs you hear about.

So what makes it so difficult to solve? If that is too broad, is there a piece to the proof that people believe, if solved, would lead to a proof?

In: Mathematics

You’re absolutely right that if you could reach every value by the inverse map, you could prove it. But that turns out to be just as hard as the original problem. The trouble is, no one has been able to find a shortcut for the Collatz map that makes it easier to reason about. There’s no obvious way to take a number and say where it ends up after 100 iterations without applying the map 100 times (or using casework mod 2^100, or something along those lines). It behaves almost like making random choices between n/2 and (3n+1)/2.

We’re not all that close to a proof, so I don’t think there’s any single piece that would get us there. At best, we have a [recent result](https://arxiv.org/pdf/1909.03562.pdf) showing that for almost all values, the Collatz map almost stays bounded (for very well defined meanings of almost). But that’s still a very long way from a full proof.