BusyBeaver(6) Is Quite Large

For overdetermined reasons, I’ve lately found the world an increasingly terrifying and depressing place. It’s gotten harder and harder to concentrate on research, or even popular science writing. Every so often, though, something breaks through that wakes my inner child, reminds me of why I fell in love with research thirty years ago, and helps me forget about the triumphantly strutting factions working to destroy everything I value.

Back in 2022, I reported an exciting advance in BusyBeaverology: namely, whereas we previously knew merely that BB(6) > 10^36,534, Pavel Kropitz managed to show that

BB(6) > ¹⁵10.

For those tuning in from home, here BB(6) is the 6^th Busy Beaver number, i.e. the maximum number of steps that a 6-state Turing machine with a {0,1} alphanet can take before halting, when run on an initially all-0 input tape. Also, the left-superscript means tetration, or iterated exponentiation: for example, ¹⁵10 means 10 to the 10 to the 10 and so on 15 times.

By comparison, last year the international “BBchallenge” team determined that BB(5) is “merely” 47,176,870 (see also Quanta magazine’s superb feature article on that milestone). So, between 5 and 6 is where the Busy Beaver function makes its leap, from the millions to beyond the bounds of observable reality.

But if you thought that was the end of the BB(6) story, think again! Eleven days ago, Tristan Sterin, who organized the BBchallenge the team, emailed to tell me that a team member with the handle “mxdys” improved the BB(6) bound yet further, to

BB(6) > ^10,000,00010

(i.e., 10 to the 10 to the 10 and so on 10 million times), with a correctness proof in Coq. Then, three days ago, Tristan wrote again to say that mxdys has improved the bound again, to

$$ BB(6) \gt ^{^{{^9}2}2}2 $$

I.e., BB(6) is at least 2 tetrated to the 2 tetrated to the 2 tetrated to the 9. So in particular, BB(6) is at least 2 pentated to the 5, where pentation is iterated tetration, i.e. the operation that is to tetration as tetration is to exponentiation, expenentiation is to multiplication, and multiplication is to addition.

Last week, when we “merely” knew that BB(6) > ^10,000,00010, I talked to a journalist who asked me to give an intuitive sense of how big such a number is. So I said, imagine you had ^10,000,00010 grains of sand. Then you could … well, uh … you could fill about ^10,000,00010 copies of the observable universe with that sand. I hope that helps people visualize it!

The journalist also asked: have these new discoveries about BB(6) caused me to rethink any broader beliefs about the Busy Beaver function? And I mean, yes and no: it was always completely within the realm of possibility that BB(6) would already be, not some puny little thing like 10^36,534, but way out in iteration land. Now that we know for sure that it is, though, maybe I ought to conjecture that the value of BB(n) becomes independent of the ZFC axioms of set theory already when n is 7 or 8 or 9, rather than when it’s 20 or 30 or whatever. (Currently, we know that BB(n) becomes independent of ZFC only when n=745.)

Unrelated Update: I’m just now returning to the US from STOC’2025 in Prague, where I saw lots of old friends and learned many interesting new things, again helping to distract me from the state of the world! Many I’ll write about some of those things in a future post. For now, though, anyone who’s interested in my STOC plenary lecture, entitled “The Status of Quantum Speedups,” can check out the PowerPoint slides here.

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