There are well defined finite integers that cannot be calculated, for example the Busy Beaver function produces them because at some point it grows faster than stacked exponentials or the Ackermann function or any other computable sequence. The Busy Beaver is a N state Turing Machine which starts with an all zero tape, and BB(N) is defined as the largest FINITE number of 1's printed on the tape AFTER it halts. Some Turing Machines never halt and just keep printing 1's forever but they don't count, it must halt.
We've known the first 4 Busy Beaver numbers for about 40 years, they are: BB(1)=1 BB(2)=4 BB(3) =6 BB(4)=13 and as of today we also know the fifth BB (5) = 2098 Nobody knows what BB(6) is and probably nobody will ever know, but we do know that it's larger than 10^10^10^10^10^10^10^10^10^10^10^10^10^10^10. A related concept called S(n) is the number of moves an n State Turing machine makes before it halts. Here are the correct values for the first 5 busy beaver machines that halt: s(1) = 1 move S(2) = 6 moves S(3) = 21 moves S(4) = 107 moves S(5) = 47,176,870 moves With Fifth Busy Beaver, Researchers Approach Computation’s Limits <https://www.quantamagazine.org/amateur-mathematicians-find-fifth-busy-beaver-turing-machine-20240702> John K Clark See what's on my new list at Extropolis <https://groups.google.com/g/extropolis> oml -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CAJPayv0WwHVJsfCZ0akR8Xy_MvbyFztKQmN4aijJ%2B0vg5Nbmow%40mail.gmail.com.
