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 2004-02-14, 16:03 #1 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 22×33×19 Posts Mind Boggling Number Mind Boggling Number. The largest number that can be written using only 3 digits is 9^9^9. Mathematician and editor Joseph S. Madachy asserts that 1)With a knowledge of the elementary properties of numbers 2) a simple desk calculator The last 10 digits of this fantastic number (and other bigger nos.) have been calculated. For the last 10 digits of 9^9^9 these have been calculated and are 2,627,177,289. Can any one give me a method with the above conditions? Note 9^9^9 is not equal to 9^81
 2004-02-14, 16:25 #2 cyrix   Jul 2003 Thuringia; Germany 2×29 Posts begin with 9 then multply with 9 (so you get max. 1 digit more) if the result has more than 10 digits, remove the first (highest) and iterate this 9^9 times (this will take a while, but it works) When your calculator has more than 11 digits to work (normaly 13) you could "optimize" this by taking a few iteration at once (multiplying with 9^3). So you have to do only 9^9/3 steps. Are there better possibilities to solve the problem? Cyrix
 2004-02-14, 22:34 #3 patdumpsite   Dec 2003 Albany, NY 2×3 Posts 9^(9^9) = 9^387420489 Thus you need only multiply 9 by itself 387420489 times. To make your calculations easier, I suppose you could keep multiplying 9 by itself until the last 10 digits started repeating themselves (which is bound to happen). I haven't given it any thought, but will this repetiton begin before we are done computing the actual value? I suspect that it might.
 2004-02-15, 00:07 #4 cyrix   Jul 2003 Thuringia; Germany 2·29 Posts The order of 9 in the multiplicative group Z(10^10)* (the group of all integers relativly prime to 10^10), which means the lowest integer p>0, for which 9^p == 1 mod (10^10), is 250,000,000 (calculated with Maple). With this knowledge you have to do "only" 387420489-250000000 iterations. cyrix
 2004-02-15, 08:40 #5 Gary Edstrom   Oct 2002 1000112 Posts Of course, if you allow the use of Donald Knuth's Arrow Notation, there is no limit to the size of number that can be represented with even just 2 digits. Since there is no up arrow on the standard keyboard, let's use "^" instead. Now, you can write the number 9^^9 in arrow noation. This can be written out as 9^(9^(9^(9^(9^(9^(9^(9^9))))))). If that isn't big enough, you could write 9^^^9. You couldn't even begin to expand it, much less comprehend its value.
 2004-02-15, 08:53 #6 michael   Dec 2003 Belgium 5×13 Posts Try to prove that 9(9[sup]9)[/sup]>((9!)!)! -michael
 2004-02-15, 11:16 #7 jinydu     Dec 2003 Hopefully Near M48 175810 Posts That reminds me. How do you obtain an approximation for the factorial of any natural number, n? I want at least the first few digits to be accurate, but avoid overflowing my calculator (which is limited to numbers < 10^100).
2004-02-15, 15:33   #8
wblipp

"William"
May 2003
New Haven

94116 Posts

Quote:
 Originally Posted by jinydu That reminds me. How do you obtain an approximation for the factorial of any natural number, n?
Stirling Series

2004-02-15, 16:36   #9
rogue

"Mark"
Apr 2003
Between here and the

22·5·17·19 Posts

Quote:
 Originally Posted by michael Try to prove that 9(9[sup]9)[/sup]>((9!)!)! -michael
I can't prove that, but I could show the opposite is true. 9[sup](99 = 9387420489 while ((9!)!)! = (362880!)!

9387420489 has fewer than 387420489 digits
362880! itself has well over a million digits. That means that (326880!)! will easily exceed 387420489 digits. I doubt I need to do any math in order for that to be obvious.

2004-02-15, 18:40   #10
wblipp

"William"
May 2003
New Haven

23·103 Posts

Quote:
 Originally Posted by Gary Edstrom Of course, if you allow the use of Donald Knuth's Arrow Notation, there is no limit to the size of number that can be represented with even just 2 digits.
I think the constraint "only three digits" should be interpretted to mean "and no other symbols, either." Then exponentiation can be shown by positioning as 99[sup]9[/sup]. If we allow non-digit symbols, then simple repetition of (x)! can turn a single 9 into an arbitrarily large number.

2004-02-15, 23:10   #11
Maybeso

Aug 2002
Portland, OR USA

4228 Posts

Quote:
 Originally Posted by wblipp I think the constraint "only three digits" should be interpretted to mean "and no other symbols, either."
And to anticipate the obvious, let's restrict it further to
"The largest number that can be written using only 3 digits, base 10, and no other symbols, is 99[sup]9[/sup].

Last fiddled with by Maybeso on 2004-02-15 at 23:10

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