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A friend and I were sitting in our cubes at work and trying to create the greatest bounded number we could using only a few characters.

We came up with $A(G,G)$, which is the Ackermann function with Graham's number $G$ as the '$M$' and '$N$' variables.

Beyond the fact that this number, though technically a bounded number, seems absolutely unquantifiable, are there larger numbers that we missed?

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Have you seen this: scottaaronson.com/writings/bignumbers.html? –  Srivatsan Oct 4 '11 at 20:54
    
You need to carefully specify what and how many symbols are allowed. For example, I have just defined $H=G!$ and propose $A(H,H)$ as larger. This is in the sense that without context, most would not recognize $A$ and $G$ this way. –  Ross Millikan Oct 4 '11 at 21:15
    
I would specify that characters that preform an operation - such as +, -, *, / would count, and characters that don't preform an operation, such as commas and parentheses, are not included. however, if one was to use a parentheses to preform an operation i.e. a(b), it would be counted. –  Christopher Rayl Oct 4 '11 at 21:23
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@SrivatsanNarayanan, that was an excellent read. –  John Gietzen Oct 4 '11 at 21:27
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In the early 1960's, as an undergraduate, I read a math newsletter that had a "Large Number Contest" - define the largest number whose definition could be typed on a postcard. There were many ingenious entries, but the hard part was not defining the entries but comparing them to see which were bigger. That's probably unsolvable. –  marty cohen Oct 8 '11 at 23:33

1 Answer 1

The language has to be specified precisely. Small differences in expressive power translate into giant differences in the size of the numbers that can be named.

A contest in 2001 for largest number generated by a C program of up to 512 characters:

http://djm.cc/bignum-results.txt

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