From: qscgz@aol.com (QSCGZ)
Subject: Re: Write the biggest number
Date: 29 Jun 2000 13:32:57 GMT
Newsgroups: sci.math
Summary: [missing]

Jim Ferry wrote:

 >M(k) = "Nest[#!&,9," ~ M(k-12) ~ "]"

 [#!&  almost expresses my thoughts...
can't we restrict to some few fundamental notation rules ?

 >...This doesn't work.  It makes A[x,y] identically equal to 9.

A[x_,y_]:=A[x-1,A[x,y-1]];A[0,y_]:=9y;A[x_,0]:=x;A[9!,9]
(56 characters , do we really need these "_"?)
I'm calculating:
A[0,y]=9y
A[1,y]=9^y
A[2,y]=9^(9^(...{y times} ^2))..)
...

 >> ...Or:
 >> Let p(1),..,p(z(n)) be the possible strings with n characters.
 >> Interprete each of the z(n) strings as a Mathematica program and run it
 >> for M(n-1) seconds. (hoping , that invalid or destructive programs won't
 >> crash the computer !)
 >> Let M(n) be the maximum of all the outputs ...
 >
 >Hmmm.  The last one is kind of a problem, even a number as small
 >as M(6) would certainly crash Mathematica. 

This is only theory. You needn't calculate the numbers in practice ;-).

 >Furthermore, I don't know how you'd implement such a thing in Mathematica.

Can't we generate files or subroutines with M. and call them ?
Well , it needn't be a M.program at all. Just create all .COM programs
of that length and call them (supposing the computer won't crash) . 
Stop after M(n-1) seconds. If it generated a new maximum , and verified
it , then take it as the new maximum.
We should require the program to include a subroutine
for deciding whether the number is bigger than the current maximum.
It's getting complicated now.

 >It's very important that any number described be implementable.  One
 >wants to avoid paradoxes.  BTW, "One plus the largest integer that
 >can be expressed in English in less than 256 characters" has less
 >than 256 characters.

that's why I included a condition on the running time.


--qscgz
==============================================================================

From: Jim Ferry <jferry@[delete_this]uiuc.edu>
Subject: Re: Write the biggest number
Date: Thu, 29 Jun 2000 10:10:07 -0500
Newsgroups: sci.math

QSCGZ wrote:
> 
> A[x_,y_]:=A[x-1,A[x,y-1]];A[0,y_]:=9y;A[x_,0]:=x;A[9!,9]
>
> (56 characters , do we really need these "_"?)
> I'm calculating:
> A[0,y]=9y
> A[1,y]=9^y
> A[2,y]=9^(9^(...{y times} ^2))..)
> ...

Yep.  Certainly an improvement of what I had.  Thus for k >= 55
we have the following candidate for M(k):

M(k) = "A[x_,y_]:=A[x-1,A[x,y-1]];A[0,y_]:=9y;A[x_,0]:=x;A[" ~
M(k-54) ~ ",9]"

Yes, the "_" are necessary:  A[x,0] = x would result in A[x,0]
being x, but A[1,0] being undefined.

It is useful to stick to Mathematica notation because it sets a standard.
When someone writes 9!! he may or may not mean 9*7*5*3*1, so it's unclear
even what M(3) should be if we're being loose.  For, say, M(100) there might
be a really big number that a few mathematicians agree is the winner, but
most think is not well-formed, followed by another which a few more agree is
well-formed, etc.

|             Jim Ferry              | Center for Simulation  |
+------------------------------------+  of Advanced Rockets   |
| http://www.uiuc.edu/ph/www/jferry/ +------------------------+
|    jferry@[delete_this]uiuc.edu    | University of Illinois |
==============================================================================

From: Jim Ferry <jferry@[delete_this]uiuc.edu>
Subject: Re: Write the biggest number
Date: Wed, 28 Jun 2000 11:29:10 -0500
Newsgroups: sci.math

I wrote:
> 
> I was thinking about a contest to write down the biggest number
> in some number of characters, using Mathematica as a convention
> for interpretation.  E.g., the best I could do in 1, 2, 4, and 60
> characters is:
> 
> 9
> 9!
> 9^9!
> A[x_,y_:9]:=A[x-1,A[x,y-1]];A[0,y_]:=9y;A[x_,0]:=A[x-1];A[9]
> 
> How about 100 characters?  I hesitate to propose such a contest.
> It would probably be too difficult to judge.

Three comments:

1)  Michael Ulm pointed out that "9! !" should win for four characters.
Mathematica interprets it as (9!)!, so it certainly beats "9^9!" (which
Mathematica evaluates as 9^(9!), rather than (9^9)!).

Define M(k) to be the maximal k-digit string.  We (tentatively) have M(1)
through M(4) as "9", "9!", "99!", and "9! !".  This yields an obvious
candidate for general M(k):

M(k) = M(k-2) ~ " !"    (where ~ is used to denote concatenation).

However, for k >= 13, this is superceded by

M(k) = "Nest[#!&,9," ~ M(k-12) ~ "]"

which gives a 9 hit with the factorial function M(k-12) times.
The next step up might be to nest the function f(x) = 9(!)^x (if
you'll pardon the notation).

And for k >= 60, we have the proposal

M(k) = "A[x_,y_:9]:=A[x-1,A[x,y-1]];A[0,y_]:=9y;A[x_,0]:=A[x-1];A[" ~
M(k-59) ~ "]"

which certainly supercedes the M(k-12) -> M(k) proposal.

2)  qscgz wrote:

> I'm not familar with Mathematica , what's the ":9" for , why have A[x-1]
> and A[9] only one index ?

The ":9" says that the second argument has a default value of 9.
Thus, A[x-1] = A[x-1,9] and A[9] = A[9,9].

I'd have rather picked freeware, but Mathematica is the only package
I'm familiar with.  I guess I could've picked C, but I like the arbitrary
size of Mathematica's numbers.

> Can't we save some characters by setting A[x_,0]:=x ?

This doesn't work.  It makes A[x,y] identically equal to 9.

> And then let's use the saved characters for a higher index in A[] ,
> e.g. A[9!,9!] .
> I think , the number of iterations is more important than the exact
> recursion-rule or the initial conditions.

Yes.  I agree.  I've incorporated this suggestion above.

>  >How about 100 characters?  I hesitate to propose such a contest.
>  >It would probably be too difficult to judge.
> 
> A[x,9] , where x the biggest 100-C character number (supposing C
> characters are needed to define such an A[] , C<=60 as we have seen ).
> Ah, wait a minute , with the definition of A[] already present , we can
> use it to get bigger numbers for x with less than C characters.
> So A[9,9] is a better candidate for x ...
> 
> What about :
> for i=1 to 9!
> A[x,y]=A[x-1,A[x,y-1]];A[0,y]:=B[y];A[x,0]:=x;
> B[x]:=A[x,x];
> next i
>
> Or: B[x,y,z]=B[x-1,B[x,y-1,z],B[x,y,z-1]]
> 
> Or:
> Let p(1),..,p(z(n)) be the possible strings with n characters.
> Interprete each of the z(n) strings as a Mathematica program and run it
> for M(n-1) seconds. (hoping , that invalid or destructive programs won't
> crash the computer !)
> Let M(n) be the maximum of all the outputs ...

Hmmm.  The last one is kind of a problem, even a number as small
as M(6) would certainly crash Mathematica.  Furthermore, I don't
know how you'd implement such a thing in Mathematica.

It's very important that any number described be implementable.  One
wants to avoid paradoxes.  BTW, "One plus the largest integer that
can be expressed in English in less than 256 characters" has less
than 256 characters.

3)  Logical Pike wrote:

> What would 9^9^9^9 be in mathematica (written the other way, of
> course)?

9^9^9^9 = 9^(9^(9^9)) both in Mathematica and in standard mathematical
usage.  I don't grok "written the other way".  Do you mean "6v6v6v6"?

|             Jim Ferry              | Center for Simulation  |
+------------------------------------+  of Advanced Rockets   |
| http://www.uiuc.edu/ph/www/jferry/ +------------------------+
|    jferry@[delete_this]uiuc.edu    | University of Illinois |