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The rec.puzzles Hall of Fame is a compilation of over 500 of the most popular puzzles that have been posted and discussed in the rec.puzzles newsgroup. In most cases a detailed solution has been provided.

Many of these puzzles also appear in Braingle's own collection.

   
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Categories : pickover : pickover.11.p

Title: Cliff Puzzle 11: The Leviathan Number
From: cliff@watson.ibm.com

If you respond to this puzzle, if possible please send me your name,
address, affiliation, e-mail address, so I can properly credit you if
you provide unique information.  PLEASE ALSO directly mail me a copy of
your response in addition to any responding you do in the newsgroup.  I
will assume it is OK to describe your answer in any article or
publication I may write in the future, with attribution to you, unless
you state otherwise.  Thanks, Cliff Pickover

      * * *


   Many interesting observations have recently been published
concerning various number theory properties of the "number of the
beast", 666.  In this new
puzzle here I ask you to consider the monstrous
"leviathan number",
a number so large as to make the number of electrons,
protons, and neutrons in the universe (10**79) pale in comparison.  (It
also makes a googol (10**100) look kind of small).

The leviathan number is defined as (10**666)!, where the "!" indicates
factorial.

1.  What are the first 6 digits of the leviathan number?  Hint:  you
need not actually compute the leviathan to determine this.  If you can
determine the first 6 digits, please carefully spell out your method.

2. Could modern supercomputers compute the leviathan, or will this
beyond the realm of humankind for the next century?

3. Even if we cannot compute the leviathan, how many other
characteristics of this number can we write down.


Solution
-------------------------

Subject: Re: Cliff Puzzle 11: The Leviathan Number (PARTIAL SPOILER)
Newsgroups: rec.puzzles
References: <1992Oct21.135208.119425@watson.ibm.com>

In article <1992Oct21.135208.119425@watson.ibm.com>, Cliff Pickover writes:

> The leviathan number is defined as (10**666)!, where the "!" indicates
> factorial.

> 1.  What are the first 6 digits of the leviathan number?

The simplest technique would be to use Stirling's formula to compute
the mantissa, i.e. frac( log(n) ) = frac( log(2*pi)/2 + log(n)/2
n*(log(n)-log(e)) ).  In our case n = 10^666, so this equals
frac( log(2*pi)/2 + 333 + 10^666*(666-log(e)) ) =
frac( log(2*pi)/2 + 10^666*(1-log(e)) ), so we'd basically need
to know something like 10 digits to the right of the decimal point
for log(2*pi)/2, and something like 700 digits for log(e) (which is
easily doable).  We then compute (1-log(e)), shift the digits 666
spaces to the left, and we're all set.

> 2. Could modern supercomputers compute the leviathan, or will this
> beyond the realm of humankind for the next century?

The number of digits is more than 10^668, and this compares
unfavorably to the number of particles in the universe.  Furthermore,
even if a googol digits could be output per second, you'd never
make it before the end of the universe.  So, I'd say it's beyond
the realm of humanity, period.

> 3. Even if we cannot compute the leviathan, how many other
> characteristics of this number can we write down.

As another puzzle, how many zeroes does it end with, and what are
the last two non-zero digits?
.qq
&EXIT
                    THIS FILE HAS BEEN RECEIVED FROM BITNET

          The file may be executable.  Before removing this header you
          must understand  what the code  will do. You must  also have
          the  appropriate intellectual  property agreements  in place
          before receiving the code into IBM.

          If you have any questions, contact your manager.


The contents of the file has been shifted right by one character.
   Filename=(none) Filetype=(none) RECFM=F LRECL=80 Records=21
The file received from the BITNET gateway begins below the next line.
------------------------------------------------------------------------
 Date:     Thu, 22 Oct 1992 07:12 EDT
 From:     
 Subject:  RE: googol!
 To:       CLIFF@YKTVMV
 Original_To:  Jnet%"CLIFF@YKTVMV"

 Hi, Cliff.

 The log10(e) comes from applying Stirling's approximation
 for the factorial: for large n, n! is approximately
 sqrt(2*pi*n)*((n/e)^n).  Substitute googol for n, take
 log10 of both sides, and recall the mantissa of the log10
 gives the digits of the original number.

 In these days of fast symbolic packages allowing exact
 computation of large factorials (though presumably not
 so large as a googol), people forget Stirling's formula.
 Until a few years ago, this was the only way to find
 factorials (albeit, only approximately) for large numbers.

 Mike





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