Googology Wiki

This wiki's URL has been migrated to the primary fandom.com domain.Read more here

READ MORE

Googology Wiki
Advertisement
Googology Wiki

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.[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 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.[4]

Tim Urban calls pentation a "power tower feeding frenzy".[5][6]

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.[7]

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

See also

Advertisement