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## #1 2007-02-20 02:37:41

ganesh
Registered: 2005-06-28
Posts: 30,265

### 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!

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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## #2 2007-02-20 04:58:25

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

### 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 2007-02-20 16:09:46

ganesh
Registered: 2005-06-28
Posts: 30,265

### 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.

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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## #4 2007-02-20 19:12:02

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

### Re: Factorials, Hyperfactorials and Superfactorials

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

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## #5 2007-02-20 19:45:38

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

### 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!

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## #6 2007-02-20 20:16:43

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

### 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)

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## #7 2007-02-21 09:15:09

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

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

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## #8 2007-02-21 16:28:21

ganesh
Registered: 2005-06-28
Posts: 30,265

### 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.

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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## #9 2016-12-03 22:34:58

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

### Re: Factorials, Hyperfactorials and Superfactorials

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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## #10 2016-12-13 04:33:27

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

### Re: Factorials, Hyperfactorials and Superfactorials

Where are these used? In Physics?

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## #11 2016-12-13 05:23:55

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

### Re: Factorials, Hyperfactorials and Superfactorials

Number theory maybe.

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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## #12 2016-12-13 15:52:47

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,910
Website

### Re: Factorials, Hyperfactorials and Superfactorials

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

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #13 2016-12-13 17:37:40

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

### Re: Factorials, Hyperfactorials and Superfactorials

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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