It has been proven that the longest-running 5-state 2-symbol Turing machine which halts is the one that halts in 47,176,870 steps. All 5-state 2-symbol Turing machines which run longer than this have been proven to run forever.

The halting theorem famously proves that there's no general algorithm that can determine whether a given computer program halts. Some programs we know halt because we can run them to completion. Others enter obvious infinite loops. But for some programs, it's hard to tell if they eventually halt or just run for a very long time. It's tantalizing to wonder where the boundary point is between simple programs whose halting status we can figure out and those programs we can't.

Research into the busy beaver problem probes and pushes that boundary. When I posted about it a year ago in P52827, there were several remaining 5-symbol 2-state Turing machines whose halting status was unknown. They were conjectured to run forever, but it was not proven. Now it's been proven, and the frontier has been pushed out to BB(6).

Homepage of the collaboration that proved it:
https://bbchallenge.org/
Announcement of the discovery:
https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237
Machine-verifiable formal proof of BB(5) = 47,176,870 in Coq:
https://github.com/ccz181078/Coq-BB5
A well-written news story on the feat (sadly behind Cloudflare):
https://www.quantamagazine.org/amateur-mathematicians-find-fifth-busy-beaver-turing-machine-20240702/

Some interesting machines at the new frontier (BB(6) and others):
https://wiki.bbchallenge.org/wiki/Cryptids

P102051
P102053
This is an achievements without a doubt. However, how viable is actually solving for higher numbers?
https://www.sligocki.com/2022/06/21/bb-6-2-t15.html
I don't think we will ever have a solution to the halting problem, and thus the team will have to continue their work manually. Which from what I'm seeing is
>reduce the number of possible machines
>get rid of a few large ones
>brute force the way through the rest

I know the community is made up of mathematicans, but I do wonder if the search for BB(6) or BB(7) will end up at least partially using something like Minecraft@Home. There is only a one line wiki page about the idea so far
https://wiki.bbchallenge.org/wiki/Accelerated_Simulator

Is there any practical use to proving the busy beavers?

P102104
Knowing BB(5) will not make anyone's number of dollars go up. As for if either of those have meaningful value, I don't know.