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

]]>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)

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)

]]>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!

]]>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.]]>

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

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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