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GNumber

Graham's number.

Graham's number is the largest "commonly known" number. It was named after a Ronald Graham, for he was the man who first calculated this number to solve an equation needed to complete a question in Ramsey theory, a question that asks how many dimesions of space would you need to have a hypercube with all of its vertices connected having an X-inside-a-square in the same plane in the same color no matter how you color the hypercube. Note that this is an upper bound and a weak one at that (the full story is available at [googology.wikia.com/wiki/Graham's_number|Googology Wiki]).

First off, I'm going to need to explain a few things if you want to know what I'm talking about.

The equation uses a system called Knuth's up arrow notation, which involves a lot of layers from multiplication up to exponents and beyond that. For now, the "^" symbol will represent an arrow.

So we have several stages of operations we will need to go through here.

First is multiplication. you know, 2*3=6. ( * is like a times symbol.)

Next you know, we have exponents. For example, 4^3=4*4*4=64. ( ^ meaning to the power of.)

And then Knuth's arrow notation, or as we'll call it, "tetration." It is as to exponents as exponents is to multiplication. For example, 3^^4=3^3^3^3 <right-to-left resolution> = 3^3^27 = 3^7625597484987 = over 3.6 trillion digits (O;o)

Now, after tetration, we have pentation (i.e. 3^^^4), hexation (i.e. 3^^^^4), and so forth, but legitimate examples of those are very few that can be written out (3^^^2 = 7625597484987 and 2^^^3 = 65536; no nontrivial cases of hexation and beyond can be written out in the observable universe), so we'll move on.

To calculate this monster, we need to hexate 3 by 3.

g1 = 3^^^^3

This is the 1st of 64 layers needed to get this number. This alone is a "tetrational tower" a "power tower of 3's 7625597484987 terms high" high.

3^^^^........(with g1 arrows in between in total).......^^^^3

This is the power of g2, or the second layer. Now imagine this done 62 MORE TIMES, all the way up to g64. this is now Graham's Number, or "G."

NOW, imagine if you were to do

G^^^^.....(With G arrows in between in total)....^^^^G

This is the story of the xkcd Number ("barely" larger than G65!).

Note that the original upper bound (until he changed it for some strange reason) was approximately g8, and a "variant" of Graham's number (the Conway-Graham Number) uses 4's instead of 3's. The new upper bound is 2^^2^^2^^9 <2^^^6, whereas the lower bound improved from 6 to 13.

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