Open problem in mathematics

A conjecture is an open problem in mathematics for which some famous dude gave heuristic arguments which indicate if the theorem is true or false.

This section groups conjectures that are famous, solved or unsolved.

They are usually conjectures that have a strong intuitive reasoning, but took a very long time to prove, despite great efforts.

The list: en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics

Given stuff like arxiv.org/pdf/2107.12475.pdf on Erdős' conjecture on powers of 2, it feels like this one will be somewhere close to computer science/Halting problem issues than number theory. Who knows. This is suggested e.g. at The Busy Beaver Competition: a historical survey by Pascal Michel.

The Collatz function is not very elegant in that the odd $3n+1$ case is always even because $n$ is odd, so it is always predictably followed by a division by two. This is not the case for the even case, where the result can be either even or odd.

A much more elegant formulation is to immediately also divide by two when the number is odd:

There are to ways in which the Collatz conjecture can fail:These are the only two options because if any sequence has an upper bound, it must sooner or later repeat an element, leading to a cycle.

We ust use the if mod notation definition as mentioned at: math.stackexchange.com/questions/4305972/what-exactly-is-a-collatz-like-problem/4773230#4773230

Described at: arxiv.org/pdf/2107.12475.pdf where a relation to the Busy beaver scale is proven, and the intuitive relation to the Collatz conjecture described. Perhaps more directly: demonstrations.wolfram.com/CollatzSequenceComputedByATuringMachine/