3 ms·
Two dozen proofs until BB(5,2) is resolved
- a_shovel 5y agohttps://en.wikipedia.org/wiki/Busy_beaver https://en.wikipedia.org/wiki/Busy_beaver At first, it seems like a magic shortcut to math. Instead of doing difficult logic stuff, just turn your problem into a binary Turing machine with n states where it halts if it finds a counterexample. Then look up BB(n,2), run it for that many steps, and if it hasn't halted by then, you've proved there's no counterexamples. Goldbach's conjecture has an n of 43, maybe even just 27. There's just two problems. One, per Wikipedia, "There is not enough computational capacity in the known part of the universe to have performed even [BB(6,2)] operations directly." And second, in order to know BB(27,2) for example, you'd have to prove which 27-state binary Turing machines halt and which ones don't. That means if Goldbach's conjecture has a 27-state representation, then you'd have to have already resolved it one way or the other to get BB(27,2). So it only lets you prove what you already know.