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Pentation refers to the 5th hyperoperation starting from addition. It is equal to $$a \uparrow\uparrow\uparrow b$$ in Knuth's up-arrow notation and since it is repeated tetration, it produces numbers that are much larger. Just a simple 2^^^3 give an amazing 65,536.

Pentation can be written in array notation as $$\{a,b,3\}$$, in chained arrow notation as $$a \rightarrow b \rightarrow 3$$ and in Hyper-E notation as E(a)1#1#b.

Pentation is less known than tetration, but there are a few googologisms employing it: 3 pentated to 3 is known as tritri, and 10 pentated to 100 is gaggol.

Sunir Shah uses the notation $$a * b$$ to indicate this function. Jonathan Bowers calls it "a to the b'th tower". Sbiis Saibian proposes $$_{b \leftarrow}a$$ in analogy to $${^{b}a}$$ for tetration, though he usually uses up-arrows.

Pentational growth rate is comparable to $$f_4(n)$$ in the fast-growing hierarchy.

A strip from the webcomic Saturday Morning Breakfast Cereal suggested the name "penetration" in humorous analogy with sexation.

Tim Urban calls pentation a "power tower feeding frenzy".

In Notation Array Notation, it is written as (a{3,3}b).

Graham, Rothschild and Spencer call the function $$f_4(n)$$ in the fast-growing hierarchy, which is faster than $$2\uparrow\uparrow\uparrow n$$ the WOW function, and corresponding growth rate wowzer.

## Examples

Here are some small examples of pentation in action:

• $$1 \uparrow\uparrow\uparrow b = 1$$
• $$a \uparrow\uparrow\uparrow 1 = a$$
• $$2 \uparrow\uparrow\uparrow 2 = 4$$
• $$2 \uparrow\uparrow\uparrow 3 = {^{^{2}2}2} = {^{4}2} = 2^{2^{2^{2}}} = 65,536$$
• $$3 \uparrow\uparrow\uparrow 2 = {^{3}3} = 3^{3^{3}} =$$ $$7,625,597,484,987$$

Here are some larger examples:

• $$3 \uparrow\uparrow\uparrow 3 = {^{^{3}3}3} = {^{7,625,597,484,987}3}$$ = tritri, a power tower of 7,625,597,484,987 threes
• $$5 \uparrow\uparrow\uparrow 2 = {^{5}5} = 5^{5^{5^{5^5}}}$$
• $$6 \uparrow\uparrow\uparrow 3 = {^{^{6}6}6}$$
• $$5 \uparrow\uparrow\uparrow 5 = {^{^{^{^{5}5}5}5}5}$$
• $$4 \uparrow\uparrow\uparrow 4 = {^{^{^{4}4}4}4}$$

## Pseudocode

Below is an example of pseudocode for pentation.

function pentation(a, b):
result := 1
repeat b times:
result := a tetrated to result
return result


## Sources

1. Really Big Numbers. Retrieved 2013-06-11.
2. Bowers, JonathanArray Notation up to Three Entries. Retrieved 2013-06-11.
3. Saibian, Sbiis3.2.3 - Ascending With Up Arrows. Retrieved 2015-03-26.
4. http://www.smbc-comics.com/?id=2615
5. Prömel, H. J.; Thumser, W.; Voigt, B. "Fast growing functions based on Ramsey theorems", Discrete Mathematics, v.95 n.1-3, p. 341-358, Dec. 1991 doi:10.1016/0012-365X(91)90346-4.
6. From 1,000,000 to Graham’s Number. Wait But Why.
7. R. Graham, B. Rothschild and J. Spencer, Ramsey Theory, 2nd edition