What is the Largest Named Number?

Date: 12/08/2004 at 20:55:28
From: April 
Subject: largest named number? 

I am doing an assignment for school, and I need to know what the 
largest named number is.  I know about billions and trillions, but 
there must be even bigger numbers than that, right?

Date: 12/20/2004 at 17:05:19
From: Doctor Best
Subject: Re: largest named number? 

Hi, April! 

Thanks for writing to Dr. Math.

You can find plenty of huge numbers on the Web.  According to the 
Guinness Book of World Records, the largest lexicographically 
accepted number is the centillion, which is 10^303 in the American 
system and 10^600 in the European system.

However, you have probably heard of the google and googleplex.  The 
google is 10^100.  The googleplex is 10^(a googol).

However, there are still higher numbers.  One is called the Moser.

  Imagine this.  If I say n[1] I mean n^n. 

  If I say n[2] I mean ((n[1])[1])[1]... where there are n [1]'s.

  If I say n[3] I mean ((n[2])[2])[2]... where there are n [2]'s.

In general, n[x] means ((n[x-1])[x-1])[x-1]... where there are n 
[x-1]s. 

Then the Moser number is 2[3]. 

  2[3]           = (2[2])[2] 
  (2[2])[2]      = ((2[1])[1])[2]
  ((2[1])[1])[2] = ((2^2)[1])[2] 
  ((2^2)[1])[2]  = (4[1])[2]
  (4[1])[2]      = (4^4)[2]
  (4^4)[2]       = 256[2]

So that means that 256 has 256 [1]'s following it, which means that 
you take x = 256 and you raise 256 to the power of itself 256 times. 
How frighteningly large.

Another number that is really large is Skewes' number.  The Skewes 
number represents a special point in prime number theory.  It is 
e^e^e^79.  But it is nowhere near the Moser.

The largest number that is named is called Graham.  Create it like 
this:

  The expression a^^b means a^a^a^a... where there are b exponent 
  signs.

  The expression a^^^b means a^^a^^a^^... where there are b double 
  exponent signs.

  The expression a^^^^b means a^^^a^^^a... where there are b triple 
  exponent signs. 

Get it?

  Now the number G1 is 3^^^^3.

  The number G2 is 3^^^^^^......3, where the number of exponent 
  arrows is equal to G1.

  The number G3 is 3^^^^^^......3, where the number of exponent 
  arrows is equal to G2.

  Continuing in this fashion, we get G64. 

This is Graham's number.  It is so large that if all the matter in the
universe were turned into paper, and you had a pen that wrote at text
size of 6.6x10^-34 metres (you can't get smaller than that), it still
would not be enough to write down the number.  G64 is used as an upper
bound to a question in Ramsey Theory.  (The actual answer is believed
to be 6.)

Feel free to write back if you still have any questions.
  
- Doctor Best, The Math Forum
  http://mathforum.org/dr.math/