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 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 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 |