## CG. Graham’s Number

Graham’s number is truly, absolutely staggering…

erratum: Graham’s number is an upper bound for a problem in Ramsey theory. We erroneously stated that the problem was eventually solved, and the actual answer was six. No, six was proven to be a lower bound.

From the Wikipedia article:

Although the solution to this problem is not yet known, Graham’s number is the smallest known upper bound. This bound was found by Graham and B. L. Rothschild (see (GR), corollary 12). They also provided the lower bound 6, adding the qualified understatement: “Clearly, there is some room for improvement here.”

1. ### rmjarvis said,

April 10, 2007 at 9:21 am

It is worth noting that Geoff Exoo claims to have improved the lower limit to 11:

http://isu.indstate.edu/ge/GEOMETRY/cubes.html

Still a far cry from Graham’s number, but higher than 6. :)

2. ### PA32R said,

April 14, 2007 at 6:09 pm

And then there are some functions that grow very fast – the busy beaver function for example. And it should be pointed out that, no matter what number is named or even recursively defined, almost every number is larger.

3. ### strauss said,

October 25, 2007 at 7:54 am

Here is an amusing, mind-blowing comic strip, sent to us by a listener: (kth operation in the sequence is k-2 arrows)

4. ### Swalkyr said,

October 29, 2007 at 3:22 am

So, how would the “hyperfive of 3 and 3” be expressed?

3^^^^^3?

5. ### strauss said,

November 2, 2007 at 9:07 pm

I typed things in wrong way round initially— it’s fixed now. The hyperfive of 3 and 3 is 3^^^3, a mere 3^^3^^3 = 3^^( 3^3^3) = 3^^ (3^27) which is about 3^^(7.5 trillion) or 3^3^3^…^3 with about 7.5 trillion 3’s— a stack of exponents 7.5 trillion high.

Now the number of particles in the universe has caused us some trouble in the past, but there are perhaps 10^80, or perhaps 10^120. Either way, the number of digits in 3^^5 = 3^3^3^3^3, a stack of 3’s just five high, is MUCH greater than the number of particles in the universe; 3^^^3 is just incomprehensibly huge.

And we’re not even up to 3^^^4.

As far as a fractional number of arrows goes, as discussed in the comic, well, ya got me there.

6. ### nelson said,

September 15, 2010 at 10:30 am

some functions that grow very fast – the busy beaver function for example. And it should be pointed out that, no matter what number is named or even recursively defined.

with regards

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7. ### robben said,

September 15, 2010 at 11:07 am

And then there are some functions that grow very fast – the busy beaver function for example. And it should be pointed out that, no matter what number is named or even recursively defined, almost every number is larger.

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