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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!

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#2 2007-02-21 03:58:25

Patrick
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Re: Factorials, Hyperfactorials and Superfactorials

Is the hyperfactorial meant to be:

? If it isn't, then I'm not sure I understand your notation


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#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)=1.2.3=1 x 4 x 27=108.


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#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)=1.2.3=1 x 4 x 27=108.

Oh well, I guess I'm turning blind.. Thanks for sharing the knowledge though smile Had never heard of hyperfactorials before!


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#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)


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#8 2007-02-22 15:28:21

ganesh
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Re: Factorials, Hyperfactorials and Superfactorials

Toast,
This page gives details of Knuth's up-arrow notation. The operation becomes much more complicated when the number of up-arrows is more.


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