**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.

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.^{[1]} Jonathan Bowers calls it "a to the b'th tower".^{[2]} Sbiis Saibian proposes \(_{b \leftarrow}a\) in analogy to \({^{b}a}\) for tetration, though he usually uses up-arrows.^{[3]}

Pentational growth rate is comperable 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.^{[4]}

Tim Urban calls pentation a "power tower feeding frenzy".^{[5]}

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

Graham, Rothschild and Spencer call the function \(2\uparrow\uparrow\uparrow n\) the *WOW function*, and corresponding growth rate *wowzer*.^{[6]}

## Examples Edit

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
- \(5 \uparrow\uparrow\uparrow 2 = {^{5}5} = 5^{5^{5^{5^5}}}\)
- \(6 \uparrow\uparrow\uparrow 3 = {^{^{6}6}6}\)
- \(5 \uparrow\uparrow\uparrow 5 = {^{^{^{^{5}5}5}5}5}\)

## Pseudocode Edit

Below is an example of pseudocode for pentation.

functionpentation(a,b):result:= 1repeatbtimes:result:=atetrated toresultreturnresult

## Sources Edit

- ↑ Template:Citation/CS1
- ↑ Template:Citation/CS1
- ↑ Template:Citation/CS1
- ↑ http://www.smbc-comics.com/?id=2615
- ↑ From 1,000,000 to Graham’s Number.
*Wait But Why*. - ↑ R. Graham, B. Rothschild and J. Spencer,
*Ramsey Theory*, 2nd edition