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.
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.
- Collatz cycle: there is a cycle that loops forever and never reaches 1
- Unbounded Collatz trajectory: there is a sequence that grows without bound without looping
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/
Articles by others on the same topic
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.