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#1 2007-02-21 01:37:41
- ganesh
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Factorials, Hyperfactorials and Superfactorials
The factorial notation must be familiar to most of you.
n! (read as n factorial) is defined as
n!=n(n-1)(n-2)(n-3)...........4 x 3 x 2 x 1.
Thus, 2!=2 x 1 = 2
3! = 3 x 2 x 1 = 6,
4! = 4 x 3 x 2 x 1 = 24
5! = 5 x 4 x 3 x 2 x 1 = 120
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 and so on.
Facotrials are useful in Combinatorics (Permutations, Combinations etc.), Probability theory, Binomial theorem, Calculus etc.
Hyperfactorial is defined as
Thus,
H(1) = 1,
H(2) = 4,
H(3) = 108 and so on.
Finally, the Superfactorial.
Clifford Pickover in his 1995 book Keys to Infinity defined the superfactorial of n as
When expressed in Knuth's up-arrow notation.
n$=n!^^n!
For example,
The function grows very rapidly and as n increases, the tower of powers increases at a very quick rate.
100$ would have more powers in the tower than a Googol! And
1000$ would have more powers in the tower than
.
These are extremely large numbers, and absolutely useless to a common man!
That is because a person may never encounter a number greater than
for most of his/her life, and certainly never ever think of anything near
unless he's/she's a mathematician!
Character is who you are when no one is looking.
#2 2007-02-21 03:58:25
- Patrick
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Re: Factorials, Hyperfactorials and Superfactorials
Is the hyperfactorial meant to be:
#3 2007-02-21 15:09:46
- ganesh
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Re: Factorials, Hyperfactorials and Superfactorials
Yes, you are correct, Patrick!
That is what it is.
I had even given examples of H(1)=1,
H(2)=4, H(3)=1.2².3³=1 x 4 x 27=108.
Character is who you are when no one is looking.
#4 2007-02-21 18:12:02
- Toast
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Re: Factorials, Hyperfactorials and Superfactorials
In Knuth's upper arrow notation, how many powers do you raise n to?
#5 2007-02-21 18:45:38
- Patrick
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Re: Factorials, Hyperfactorials and Superfactorials
ganesh wrote:
Yes, you are correct, Patrick!
That is what it is.
I had even given examples of H(1)=1,
H(2)=4, H(3)=1.2².3³=1 x 4 x 27=108.
Oh well, I guess I'm turning blind.. Thanks for sharing the knowledge though 
Had never heard of hyperfactorials before!
#6 2007-02-21 19:16:43
- Zhylliolom
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Re: Factorials, Hyperfactorials and Superfactorials
Toast wrote:
In Knuth's upper arrow notation, how many powers do you raise n to?
means raise a to itself n-1 times. For example,
Then basically there is a "tower" of n a's.
I can post more on this notation once I figure out how to make the LaTeX work here. (does someone know how to do underbraces in this forum? I can't see you get it to work)
#7 2007-02-22 08:15:09
- Patrick
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Re: Factorials, Hyperfactorials and Superfactorials
Zhylliolom wrote:
Toast wrote:
In Knuth's upper arrow notation, how many powers do you raise n to?
means raise a to itself n-1 times. For example,
Then basically there is a "tower" of n a's.
I can post more on this notation once I figure out how to make the LaTeX work here. (does someone know how to do underbraces in this forum? I can't see you get it to work)
Last edited by Patrick (2007-02-22 08:15:31)