TLDR; As BB(n) gets larger, they can encode more random walk style problems that have a stop condition related to the position of the random walk. Proving that such a condition is unlikely may be easy, but proving it never occurs is very difficult.
TLDR; As BB(n) gets larger, they can encode more random walk style problems that have a stop condition related to the position of the random walk. Proving that such a condition is unlikely may be easy, but proving it never occurs is very difficult.
Way worse than just very difficult:
- “Avoid the Collatz Conjecture at All Costs!” (Math Kook) https://www.youtube.com/watch?v=TxBRcwkRjmc
- “Experienced mathematicians warn up-and-comers to stay away from the Collatz conjecture. It’s a siren song, they say: Fall under its trance and you may never do meaningful work again.” https://www.quantamagazine.org/mathematician-proves-huge-res...
- "Mathematics is not yet ready for such problems [as Collatz].” (Paul Erdős)
Iirc if you change the numerical values of the collatz problem some instances are undecidable.
A generalized Collatz problem ((mx + b mod n) instead of 3x+1 in Z) is undecidable.