This looks simple, but no one expects it to be solved in the near future. So Collatz(-like) problems are intimately connected to the halting problem and in this sense represent the "simplest" type of problem (statement) in the Cryptid/complexity hierarchy we cannot (yet?) solve. Other ((closer to the) real world) problems are higher in this hierarchy (require more states to describe: https://wiki.bbchallenge.org/wiki/Cryptids#Larger_Cryptids), and solutions/increased understanding of the Collatz problems could assist in their solution.
Thank you for giving me an answer that nobody else seems to know. I have no idea if you're correct, since i haven't studied the halting problem, although i understand the general concept.
The conjecture itself might not be as important but the tools we have to discover and use to prove this conjecture would be of paramount importance, as they'd teach us about the nature of mathematics, numbers, and more.
Many unsolved problems in mathematics are ones that are not easily understood by a lay person. The Collatz conjecture has captured hearts and minds at least in part because it is easy to understand the statement of. Perhaps proving that pi is normal would be another one.
But they are few and far between. In a sense, the reason it’s interesting is that we don’t know how to solve it.
It's a curiosity, a rabbit hole, and not likely to have practical applications. It's one of the math things I toy with from time to time, like factoring large coprimes.
To me it’s hard to believe that fundamental knowledge about computation would have no practical applications. That depends on “practical” of course. Elliptic curves or group theory were hardly of any use for medieval dirt slappers.
Finding the value for machines with 2 symbols and 6 (or more) states will require solving, among other things, the problem of whether the following Collatz-like program ("Antihydra": https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html, https://wiki.bbchallenge.org/wiki/Cryptids, https://wiki.bbchallenge.org/wiki/Antihydra) halts or not, because there exists such a machine that exhibits this behavior:
This looks simple, but no one expects it to be solved in the near future. So Collatz(-like) problems are intimately connected to the halting problem and in this sense represent the "simplest" type of problem (statement) in the Cryptid/complexity hierarchy we cannot (yet?) solve. Other ((closer to the) real world) problems are higher in this hierarchy (require more states to describe: https://wiki.bbchallenge.org/wiki/Cryptids#Larger_Cryptids), and solutions/increased understanding of the Collatz problems could assist in their solution.