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Old 2004-02-14, 16:03   #1
mfgoode
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Default 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
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Old 2004-02-14, 16:25   #2
cyrix
 
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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
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Old 2004-02-14, 22:34   #3
patdumpsite
 
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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.
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Old 2004-02-15, 00:07   #4
cyrix
 
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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.

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Old 2004-02-15, 08:40   #5
Gary Edstrom
 
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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.
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Old 2004-02-15, 08:53   #6
michael
 
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Try to prove that 9(9[sup]9)[/sup]>((9!)!)!


-michael
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Old 2004-02-15, 11:16   #7
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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).
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Old 2004-02-15, 15:33   #8
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Quote:
Originally Posted by jinydu
That reminds me. How do you obtain an approximation for the factorial of any natural number, n?
Stirling Series
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Old 2004-02-15, 16:36   #9
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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.
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Old 2004-02-15, 18:40   #10
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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.
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Old 2004-02-15, 23:10   #11
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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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