Comment by hxtk

The algorithm for BB(N) is (with some hand waving) “for each N-state Turing machine, check if it halts on an initially empty tape. If it does not, skip it. If it does, run it until it halts and count the number of steps, k. BB(N) is the maximum value of k you find.”

The problem is that the naive algorithm above requires an oracle for the Halting Problem, which every CS student learns is impossible in general for a computational model that isn’t more powerful than a Turing Machine.

So if a function can be expressed in terms of a Turing machine, busy beaver has to grow faster because eventually for some value of N, you can just encode the candidate function as a Turing machine and have Busy Beaver simulate it.

Busy Beaver hunters on the low end (trying to find the next number) have to exhaustively prove whether each Turing machine eventually halts or not at that number of states so that they can evaluate the longest running one with finite execution.