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I remember reading in RPF's biographical book, "Surely you're joking Mr. Feynman" that he used to have timed contest about writing down the biggest number using standard symbols.

Instead of the time restriction, I was wondering what if we restrict the number of characters, to say $N=10$. What is the largest number we can write using $N$ characters using numbers and standard functions known to most mathematicians? Of course I think the real difficulty would be in proving the said number is the largest. What is the largest number you can think of with $N=10$?

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Everything hinges on what you mean by "standard functions known to most mathematicians," as well as what characters you're allowing (do parentheses count extra?) and what shorthand you're allowing. If B is valid shorthand for the busy beaver function then I think it will be hard to beat BBBBBBBBB9. This number is pointlessly large. –  Qiaochu Yuan Nov 27 '10 at 1:44
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Scott Aaronson has written an entertaining essay on this very subject: Who Can Name the Bigger Number? –  Rahul Nov 27 '10 at 2:03
    
If you kept the range of allowed functions very small, it could be an interesting exercise. Say only absolute basic operators, brackets allowed free, as I don't see why order of operator precedence should interfere as that is essentially arbitary: +, -, *, /, ln, e, i. Perhaps allowing ln to be counted as a single character, abbreviated as l - the inclusion of i is not strictly necessary as it can be built (-1)^(1/2), and I omitted any root for the same reason, but it's sort of elementary. –  Orbling Nov 27 '10 at 2:27
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1 Answer

Well, just for fun I thought I'd try to answer the question with the operations restricted to:

  • nullary operations $\{0,1,2,\ldots,9\}$ and
  • binary operation $*$ (i.e. multiplication).

It turns out that

9999999999

is the greatest in this case. The closest runner-up is:

99999*9999 = 999890001

We can make some simplifying assumptions: (a) all 10 slots are to be used (otherwise, we just add in another 9 at the end) and (b) any of $0,1,2\ldots,8$ is better off being replaced by a $9$.

This exercise didn't seem all that satisfying really, there's so many sequences that result in syntax errors. E.g.:

*999*99999
999999999*
999999**99

In my opinion, it gets even less satisfying if you keep going. E.g.

  • Is 1/0 allowed?
  • Does exponentation require a character? E.g. 2^6 vs. $2^6$.
  • Do you require suitable bracketing? When? 1/2^{-6} vs. 1/2^-6. What about 9^9^9?
  • How many characters does $\log$ require?
  • The result varies on human factors -- e.g. what number system is used (base 10?), how many characters it takes to write a function, etc.
  • And, of course, there's the inevitable question of self-referential sequences (which might be difficult to arrange for 10 characters).
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Who parses 9^9^9 as (9^9)^9 anyway? –  J. M. Nov 27 '10 at 5:22
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