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| − | '''Pentation |
+ | '''Pentation''' refers to the 5th [[Hyper operator|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 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 |
+ | Pentation is less known than tetration, but there are a few [[googologism]]s 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.<ref>{{cite web|url=http://c2.com/cgi/wiki?ReallyBigNumbers|title=Really Big Numbers|accessdate=2013-06-11}}</ref> [[Jonathan Bowers]] calls it "a to the b'th tower".<ref>{{cite web|first=Jonathan|last=Bowers|authorlink=Jonathan Bowers|url=http://www.polytope.net/hedrondude/trientrical.htm|title=Array Notation up to Three Entries|accessdate=2013-06-11}}</ref> [[Sbiis Saibian]] proposes \(_{b \leftarrow}a\) in analogy to \({^{b}a}\) for tetration, though he usually uses up-arrows.<ref>{{cite web|first=Sbiis|last=Saibian|authorlink=Sbiis Saibian|url=https://sites.google.com/site/largenumbers/home/3-2/knuth|title=3.2. |
+ | Sunir Shah uses the notation \(a * b\) to indicate this function.<ref>{{cite web|url=http://c2.com/cgi/wiki?ReallyBigNumbers|title=Really Big Numbers|accessdate=2013-06-11}}</ref> [[Jonathan Bowers]] calls it "a to the b'th tower".<ref>{{cite web|first=Jonathan|last=Bowers|authorlink=Jonathan Bowers|url=http://www.polytope.net/hedrondude/trientrical.htm|title=Array Notation up to Three Entries|accessdate=2013-06-11}}</ref> [[Sbiis Saibian]] proposes \(_{b \leftarrow}a\) in analogy to \({^{b}a}\) for tetration, though he usually uses up-arrows.<ref>{{cite web|first=Sbiis|last=Saibian|authorlink=Sbiis Saibian|url=https://sites.google.com/site/largenumbers/home/3-2/knuth|title=3.2.3 - Ascending With Up Arrows|accessdate=2015-03-26}}</ref> |
| − | Pentational growth rate is |
+ | Pentational growth rate is comparable to \(f_4(n)\) in the [[fast-growing hierarchy]]. |
| − | A strip from the webcomic ''{{w|Saturday Morning Breakfast Cereal}}'' suggested the name " |
+ | A strip from the webcomic ''{{w|Saturday Morning Breakfast Cereal}}'' suggested the name "penetration" in humorous analogy with [[sexation]].<ref>http://www.smbc-comics.com/?id=2615</ref> |
| − | Tim Urban calls pentation a "power tower feeding frenzy".<ref>[http://waitbutwhy.com/2014/11/1000000-grahams-number.html From 1,000,000 to Graham’s Number]. ''Wait But Why''.</ref> |
+ | Tim Urban calls pentation a "power tower feeding frenzy".<ref>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}}.</ref><ref>[http://waitbutwhy.com/2014/11/1000000-grahams-number.html From 1,000,000 to Graham’s Number]. ''Wait But Why''.</ref> |
| + | In [[Notation Array Notation]], it is written as (a{3,3}b). |
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| + | 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|WOW function]]'', and corresponding growth rate ''wowzer''.<ref>R. Graham, B. Rothschild and J. Spencer, ''Ramsey Theory'', 2nd edition</ref> |
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| + | |||
| ⚫ | |||
Here are some small examples of pentation in action: |
Here are some small examples of pentation in action: |
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*\(2 \uparrow\uparrow\uparrow 2 = 4\) |
*\(2 \uparrow\uparrow\uparrow 2 = 4\) |
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*\(2 \uparrow\uparrow\uparrow 3 = {^{^{2}2}2} = {^{4}2} = 2^{2^{2^{2}}} = 65,536\) |
*\(2 \uparrow\uparrow\uparrow 3 = {^{^{2}2}2} = {^{4}2} = 2^{2^{2^{2}}} = 65,536\) |
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| − | *\(3 \uparrow\uparrow\uparrow 2 = {^{3}3} = 3^{3^{3}} =\) {{mathlink|7,625,597,484,987}} |
+ | *\(3 \uparrow\uparrow\uparrow 2 = {^{3}3} = 3^{3^{3}} =\) {{mathlink|7625597484987|7,625,597,484,987}} |
Here are some larger examples: |
Here are some larger examples: |
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*\(6 \uparrow\uparrow\uparrow 3 = {^{^{6}6}6}\) |
*\(6 \uparrow\uparrow\uparrow 3 = {^{^{6}6}6}\) |
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*\(5 \uparrow\uparrow\uparrow 5 = {^{^{^{^{5}5}5}5}5}\) |
*\(5 \uparrow\uparrow\uparrow 5 = {^{^{^{^{5}5}5}5}5}\) |
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| + | *\(4 \uparrow\uparrow\uparrow 4 = {^{^{^{4}4}4}4}\) |
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| − | + | == Pseudocode == |
|
| − | Below is an example of pseudocode for pentation. |
+ | Below is an example of [https://en.wikipedia.org/wiki/Pseudocode pseudocode] for pentation. |
'''function''' pentation(''a'', ''b''): |
'''function''' pentation(''a'', ''b''): |
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'''return''' ''result'' |
'''return''' ''result'' |
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| − | + | == Sources == |
|
<references /> |
<references /> |
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| − | + | == See also == |
|
| − | {{ExtendedOps}} |
+ | {{ExtendedOps}}[[ja:ペンテーション]] |
| − | |||
| − | [[ja:ペンテーション]] |
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[[Category:Functions]] |
[[Category:Functions]] |
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[[Category:Binary operators]] |
[[Category:Binary operators]] |
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| + | [[Category:Hyper operators]] |
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Revision as of 11:17, 4 July 2021
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
- ↑ Really Big Numbers. Retrieved 2013-06-11.
- ↑ Bowers, Jonathan. Array Notation up to Three Entries. Retrieved 2013-06-11.
- ↑ Saibian, Sbiis. 3.2.3 - Ascending With Up Arrows. Retrieved 2015-03-26.
- ↑ http://www.smbc-comics.com/?id=2615
- ↑ 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.
- ↑ From 1,000,000 to Graham’s Number. Wait But Why.
- ↑ R. Graham, B. Rothschild and J. Spencer, Ramsey Theory, 2nd edition
See also
Main article: hyper operator
Hyper-operators: successor · addition · multiplication · exponentiation · tetration · pentation · hexation · heptation · octation · enneation · decation (un/doe/tre) · vigintation (doe) · trigintation · centation (tri) · docentation · myriation · circulationBowers' 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...