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 case is always even because 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.
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/

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The Collatz conjecture, also known as the 3n + 1 conjecture, is a famous unsolved problem in mathematics that deals with sequences defined in a particular way. The conjecture can be described as follows: 1. Take any positive integer \( n \). 2. If \( n \) is even, divide it by 2. 3. If \( n \) is odd, multiply it by 3 and add 1.