RE: range of atoms
> -----Original Message-----
> From: a.tammer at hetnet.nl [mailto:a.tammer at hetnet.nl]
> Does the chess-example include real-games or
> all kinds of theoretically possible combinations,
> in real chess allowed or not allowed, or sometimes
> even constructed beyond in real games obtainable
> positions of pieces? 
)
Looking in the _Penguin Dictionary of Curious and Interesting Numbers_ by
David Wells, the second to the last entry is Skewes number:
10^(10^(10^34)) = 10^3400
You can estimate the number of primes less than n using the integral of
n/log n from 0 to n. This starts out as an over estimate, but switches
between that and under estimates an infinite number of times. It's been
proved that the first switch occurs before n reaches Skewes number. The
text states that, "At the time [1933] this was an extraordinarily large
number." Later, Hardy figured that it was the largest number that served
any real purpose, and that if you had a chess game where all the particles
of the universe were pieces (~ 10^80 - 10^87), and a move were the
interchange of any two particles, where the game terminated after the same
positions recured three times, the number of possible games would be Skewes
number.
The largest number listed in the dictionary, however, is Graham's number:
^^ ^^
3||...||3
Those are really arrows pointing up. Its a special notation created by
Donald Knuth. This number is in the Guinness book of records (and was
featured in Scientific American), and is thought to be an upper bound for a
combinatorics problem in Ramsey theory (some experts in Ramsey theory think
the actual answer may be as low as 6).
You just can't express this number in terms of normal powers (not enough
ink/electrons/whatever):
^
3|3 = 3^3, but
^^
3||3 = 3^(3^3) = 3^27 - 7,625,597,484,987
But wait:
^^^ ^^ ^^ ^^
3|||3 = 3||(3||3) = 3||(7,625,597,484,987)
I think you can see where this goes. Consider the number
^^^ ^^^ ^^^^
3|||...|||3 in which there are 3||||3 arrows, and call this g1.
Now contstruct g2 where there are g1 arrows, g3 which has as g2 arrows, and
so forth until you get to g63. This is Graham's number.
Matt Lewis
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