Busy Beaver 96) is really quite large

[John Clark]

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. 

BB(1)=1
BB(2)=4
BB(3) =6
BB(4)=13

BB (5) = 2098  

Sometimes Busy beaver numbers are defined slightly differently is the maximum number of steps an in-state Turing machine makes before it halts then the first five busy beaver numbers are: 

• Σ(1) = 1
• Σ(2) = 6
• Σ(3) = 21
• Σ(4) = 107
• Σ(5) = 47,176,870

We know that at some point the Busy Beaver function starts growing faster than any computable function; nobody knows exactly when that happens but it's sometime before BB(643), probably much before. There have been some very recent developments, in regard to BB(6), we still don't know what that number is but we do have a better idea of what its lower bound is then we had a month ago.  Before I can say what that is I have to say something about notation. Everybody knows that 10^15 means 10*10*10  15 times, but in this new notation 15^10 means a tower of iterated exponentiations 10 to the 10 to the 10  ...15 times.

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So the new lower limit of the sixth busy beaver number is: 

BB(6)> 9^2^2^2

I don't know this is a fact but I wouldn't be surprised if this was the largest finite number ever to come up in a mathematical investigation and not just in a contest to think of big numbers. 

John K Clark    See what's on my new list at  Extropolis

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