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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 34,969

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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**Patrick****Real Member**- Registered: 2006-02-24
- Posts: 1,005

Is the hyperfactorial meant to be:

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 34,969

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.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Toast****Real Member**- Registered: 2006-10-08
- Posts: 1,321

In Knuth's upper arrow notation, how many powers do you raise n to?

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**Patrick****Real Member**- Registered: 2006-02-24
- Posts: 1,005

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!

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**Zhylliolom****Real Member**- Registered: 2005-09-05
- Posts: 412

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)

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**Patrick****Real Member**- Registered: 2006-02-24
- Posts: 1,005

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 (2007-02-21 09:15:31)*

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 34,969

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.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**iamaditya****Member**- From: Planet Mars
- Registered: 2016-11-15
- Posts: 821

Neil Sloane and Simon Plouffe defined a superfactorial in The Encyclopedia of Integer Sequences (Academic Press, 1995) to be the product of the first n factorials. So the superfactorial of 4 is

sf(4)=4!*3!*2!*1!=288 and

sf(n)=n!*(n-1)!*(n-2)!..........2!*1!

Alternative definition

Clifford Pickover in his 1995 book Keys to Infinity used a new notation, n$, to define the superfactorial

x$=(x!)^(x!)^(x!)^(x!).....{x! times}

Source:Wikipedia

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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

Where are these used? In Physics?

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Number theory maybe.

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In Knuth and Graham's book, Concrete Mathematics, the authors use notations like falling factorials for discrete mathematics.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

That is true since Knuth invented the arrow notation.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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