Pentation or quintation refers to the function \(a \uparrow\uparrow\uparrow b\), where Arrow Notation is used. It produces numbers very much larger than those produced by tetration.[1]
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 its tetrational cousin, 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.[2] Jonathan Bowers calls it "a to the b'th tower".[3] Sbiis Saibian proposes \(_{b \leftarrow}a\) in analogy to \({^{b}a}\) for tetration, though he usually uses up-arrows.[4]
Pentational growth rate is equivalent 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]
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
- \(5 \uparrow\uparrow\uparrow 2 = {^{5}5} = 5^{5^{5^{5^5}}}\)
- \(6 \uparrow\uparrow\uparrow 3 = {^{^{6}6}6}\)
- \(5 \uparrow\uparrow\uparrow 5 = {^{^{^{^{5}5}5}5}5}\)
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
- ↑ Bowers, Jonathan. Exploding Array Function. Retrieved 2013-06-11.
- ↑ Really Big Numbers. Retrieved 2013-06-11.
- ↑ Bowers, Jonathan. Array Notation up to Three Entries. Retrieved 2013-06-11.
- ↑ Saibian, Sbiis. 3.2.5 - Ascending With Up Arrows. Retrieved 2013-06-11.
- ↑ http://www.smbc-comics.com/?id=2615
- ↑ From 1,000,000 to Graham’s Number. Wait But Why.
See also
Bowers' extensions: expansion · multiexpansion · powerexpansion · expandotetration · explosion (multi/power/tetra) · detonation · pentonation
Saibian's extensions: hexonation · heptonation · octonation · ennonation · deconation
Tiaokhiao's extensions: megotion (multi/power/tetra) · megoexpansion (multi/power/tetra) · megoexplosion · megodetonation · gigotion (expand/explod/deto) · terotion · more...