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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 \(2 \uparrow\uparrow\uparrow 3\) 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] Reuben Goodstein mentions that \(_{b}a\) would stand for pentation, analogous to how \(^{b}a\) would stand for tetration[3]; likewise Sbiis Saibian proposes \(_{b \leftarrow}a\) in analogy to \({^{b}a}\) for tetration, though he usually uses up-arrows.[4]

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.[5]

Tim Urban calls pentation a "power tower feeding frenzy".[6][7]

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

In hyperlicious function, it is equal to \(h_5(a,b)\).

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.[8]

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
  • \(5 \uparrow\uparrow\uparrow 2 = {^{5}5} = 5^{5^{5^{5^5}}}\)
  • \(6 \uparrow\uparrow\uparrow 3 = {^{^{6}6}6}\)
  • \(4 \uparrow\uparrow\uparrow 4 = {^{^{^{4}4}4}4}\)
  • \(5 \uparrow\uparrow\uparrow 5 = {^{^{^{^{5}5}5}5}5}\)

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. doi:10.2307/2266486
  4. Saibian, Sbiis3.2.3 - Ascending With Up Arrows. Retrieved 2015-03-26.
  5. http://www.smbc-comics.com/?id=2615
  6. 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.
  7. From 1,000,000 to Graham’s Number. Wait But Why.
  8. R. Graham, B. Rothschild and J. Spencer, Ramsey Theory, 2nd edition

See also[]

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