You can use the digits 1,2,and 3 once only and any mathematical symbols you are aware of, but no symbol is to be used more than once. The challenge is to see if you can make the largest numbers.

Here are some numbers to set the ball rolling: 321, 21to the third power, (3/.1)to the second power.

(levik: I guess this is more of a competition)

Nick Reed, in his post (cid=966), introduces into his solution the use of the symbol for exponential notation. x E+n is equivalent to x*10ⁿ. Nick also made use of inverson by adding a decimal point in front of the 1 and using the negative sign for the power:
           .x^-n = (1/(10 * x))^-n = (10 * x)ⁿ = 10ⁿ * xⁿ

Erik O., in his post (cid=15142), mentions tetration, pentation and hexation. Unfortunately, in his example he uses the iteration of the asterisk, which I understand is why his solution does not fall within the solution guidelines. For pentation and hexation, I do not know if either can be represented without an iteration of a known symbol, yet, with tetration, like exponentation, superscription can be used. The difference between the two is where exponetation uses the superscription following the number/variable -- xⁿ --, tetration uses the superscription preceding it -- ⁿx. 

xⁿ is equivalent to x*x*x...*x, where x is iterated n times.
eg., 2³ = 2*2*2 = 8
      4³ = 4*4*4 = 64

ⁿx is equivalent to (x^(^{x^}(^{x^}(...(^{x^x}...))))), again, where x is iterated n times.
eg., ³2 = 2^(2^2) = 16
      ³4= 4^(4^4)  =  1.3407807929942597099574024998206e+154

As just demonstrated, tetration can produce very large numbers.

Also used by Nick as a symbol to be used, yet first offered in this problem's posts by Happy (cid=352) is the factorial, x!.  Another special function, and capable of producing larger numbers, is the superfactorial x$. The superfactorial, x$, as defined by Pickover (1995), is equal to ⁿn such that n is x! -- (I have used tetration notation here as I found it easier than to show the iteration of the exponentation.  One should also note, the actual Pickover's symbol used is not a dollar sign, but an exclamation point with an S superimposed.)

In addition to the superfactorial, there is the hyperfactorial.  The hyperfactorial produces large numbers.  Though smaller than the superfactorial, these numbers are larger than those produced by the factorial.  The common notation for the hyperfactorial is
H(n), and is defined for n as 1¹*2²*3³...*nⁿ.

Another factorial-type function is the primorial.  The general notation for this function is p_n#, where p_n is the nth prime. The primorial, then, is given as 2*3*5*7*11*13...*n.

I offer the following as one of the large numbers that can be represented with only using numbers 1, 2, and 3, once, and no iteration of any other symbols. As the factorial has second notation that does not require the exclamation point, the use of both the factorial and superfactorial can be used.  The factorial's older notation can be used -- a half box with a left and bottom border: |_. But as this symbol is not available in this font, I shall use the ! here instead. 

^{{H([1E+3!]#)}$}2

I have used four different symbols for grouping mathematical expressions: the parenthesis, square brackets, curly brackets (also known as braces) and the vinculum (a horizontal line placed above to form a unit). As I am using the hyperfactorial function which uses parenthesis, I use these other symbols for grouping. 

I do not know how large this value is.  Can anyone tell me if it is smaller or larger than one googleplex? 

Edited on July 20, 2006, 10:43 pm

Posted by Dej Mar on 2006-07-19 06:52:46