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{{Too technical|date=January 2023}}{{Short description|Arithmetic operation}}
[[File:Pentation.jpg|thumb|The first three values of the expression ''x''[5]2. The value of 3[5]2 is {{val|7625597484987}}; values for higher ''x'', such as 4[5]2, which is about 2.361 × 10<sup>8.072 × 10<sup>153</sup></sup> are much too large to appear on the graph.]]
In [[mathematics]], '''pentation''' (or '''hyper-5''') is the fifth [[hyperoperation]]. Pentation is defined to be repeated [[tetration]], similarly to how tetration is repeated [[exponentiation]], exponentiation is repeated [[multiplication]], and multiplication is repeated [[addition]]. The concept of "pentation" was named by English mathematician [[Reuben Goodstein]] in 1947, when he came up with the naming scheme for hyperoperations.

The number ''a'' pentated to the number ''b'' is defined as ''a'' tetrated to itself ''b - 1'' times. This may variously be denoted as <math>a[5]b</math>, <math>a\uparrow\uparrow\uparrow b</math>, <math>a\uparrow^3 b</math>, <math>a\to b\to 3</math>, or <math>{_{b}a}</math>, depending on one's choice of notation. 

For example, 2 pentated to the 2 is 2 tetrated to the 2, or 2 raised to the power of 2, which is <math>2^2 = 4</math>. As another example, 2 pentated to the 3 is 2 tetrated to the result of 2 tetrated to the 2. Since 2 tetrated to the 2 is 4, 2 pentated to the 3 is 2 tetrated to the 4, which is <math>2^{2^{2^2}} = 65536</math>.

Based on this definition, pentation is only defined when ''a'' and ''b'' are both [[positive integers]].

== Definition ==
Pentation is the next [[hyperoperation]] (infinite [[sequence]] of arithmetic operations, based on the previous one each time) after [[tetration]] and before hexation. It is defined as [[iterated]] (repeated) tetration (assuming right-associativity). This is similar to as tetration is iterated right-associative [[exponentiation]].<ref>{{citation
 | last = Perstein | first = Millard H.
 | date = June 1961
 | doi = 10.1145/367766.368160
 | issue = 6
 | journal = [[Communications of the ACM]]
 | page = 344
 | title = Algorithm 93: General Order Arithmetic
 | volume = 5| s2cid = 581764
 | doi-access = free
 }}.</ref> It is a [[binary operation]] defined with two numbers ''a'' and ''b'', where ''a'' is tetrated to itself ''b − 1'' times. 

The type of hyperoperation is typically denoted by a number in brackets, []. For instance, using [[hyperoperation]] notation for pentation and tetration,  <math>2[5]3</math> means {{not a typo|tetrating}} 2 to itself 2 times, or <math>2[4](2[4]2)</math>. This can then be reduced to <math>2[4](2^2)=2[4]4=2^{2^{2^2}}=2^{2^4}=2^{16}=65,536.</math>

==Etymology==
The word "pentation" was coined by [[Reuben Goodstein]] in 1947 from the roots [[penta-]] (five) and [[iterated function|iteration]]. It is part of his general naming scheme for [[hyperoperation]]s.<ref>{{citation
 | last = Goodstein | first = R. L. | authorlink = Reuben Goodstein
 | journal = [[The Journal of Symbolic Logic]]
 | mr = 0022537
 | pages = 123–129
 | title = Transfinite ordinals in recursive number theory
 | volume = 12
 | issue = 4 | year = 1947| doi = 10.2307/2266486 | jstor = 2266486 | s2cid = 1318943 }}.</ref>

==Notation==
There is little consensus on the notation for pentation; as such, there are many different ways to write the operation. However, some are more used than others, and some have clear advantages or disadvantages compared to others.

*Pentation can be written as a [[hyperoperation]] as <math>a[5]b</math>. In this format, <math>a[3]b</math> may be interpreted as the result of [[Iterated function|repeatedly applying]] the function <math>x\mapsto a[2]x</math>, for <math>b</math> repetitions, starting from the number&nbsp;1. Analogously, <math>a[4]b</math>, tetration, represents the value obtained by repeatedly applying the function <math>x\mapsto a[3]x</math>, for <math>b</math> repetitions, starting from the number&nbsp;1, and the pentation <math>a[5]b</math> represents the value obtained by repeatedly applying the function <math>x\mapsto a[4]x</math>, for <math>b</math> repetitions, starting from the number&nbsp;1.<ref>{{citation
 | last = Knuth | first = D. E. | author-link = Donald Knuth
 | doi = 10.1126/science.194.4271.1235
 | journal = [[Science (journal)|Science]]
 | pages = 1235–1242
 | title = Mathematics and computer science: Coping with finiteness
 | volume = 194
 | year = 1976
 | pmid=17797067
 | issue=4271| bibcode = 1976Sci...194.1235K | s2cid = 1690489 }}.</ref><ref>{{citation
 | last1 = Blakley | first1 = G. R.
 | last2 = Borosh | first2 = I.
 | doi = 10.1016/0001-8708(79)90052-5
 | issue = 2
 | journal = [[Advances in Mathematics]]
 | mr = 549780
 | pages = 109–136
 | title = Knuth's iterated powers
 | volume = 34
 | year = 1979| doi-access = 
 }}.</ref> This will be the notation used in the rest of the article.

*In [[Knuth's up-arrow notation]], <math>a[5]b</math> is represented as <math>a \uparrow \uparrow \uparrow b</math> or <math>a \uparrow^{3}b</math>. In this notation, <math>a\uparrow b</math> represents the exponentiation function <math>a^b</math> and <math>a\uparrow \uparrow b</math> represents tetration. The operation can be easily adapted for hexation by adding another arrow.
*In [[Conway chained arrow notation]], <math>a[5]b = a\rightarrow b\rightarrow 3</math>.<ref>{{citation|title=The Book of Numbers|first1=John Horton|last1=Conway|author1-link=John Horton Conway|first2=Richard|last2=Guy|author2-link=Richard K. Guy|publisher=Springer|year=1996|isbn=9780387979939|page=61|url=https://books.google.com/books?id=0--3rcO7dMYC&pg=PA61}}.</ref>
*Another proposed notation is <math>{_{b}a}</math>, though this is not extensible to higher hyperoperations.<ref>{{Cite web |title=Tetration.org - Tetration |url=http://www.tetration.org/Tetration/index.html |access-date=2022-09-12 |website=www.tetration.org}}</ref>

== Examples ==
The values of the pentation function may also be obtained from the values in the fourth row of the table of values of a variant of the [[Ackermann function]]: if <math>A(n,m)</math> is defined by the Ackermann recurrence <math>A(m-1,A(m,n-1))</math> with the initial conditions <math>A(1,n)=an</math> and <math>A(m,1)=a</math>, then <math>a[5]b=A(4,b)</math>.<ref>{{citation
 | last = Nambiar | first = K. K.
 | doi = 10.1016/0893-9659(95)00084-4
 | issue = 6
 | journal = Applied Mathematics Letters
 | mr = 1368037
 | pages = 51–53
 | title = Ackermann functions and transfinite ordinals
 | volume = 8
 | year = 1995| doi-access = free
 | citeseerx = 10.1.1.563.4668
 }}.</ref>

As tetration, its base operation, has not been extended to non-integer heights, pentation <math>a[5]b</math> is currently only defined for integer values of ''a'' and ''b'' where ''a'' > 0 and ''b'' ≥ −2, and a few other integer values which ''may'' be uniquely defined. As with all hyperoperations of order 3 ([[exponentiation]]) and higher, pentation has the following [[Triviality (mathematics)|trivial]] cases (identities) which holds for all values of ''a'' and ''b'' within its domain:

* <math>1[5]b = 1</math>
* <math>a[5]1 = a</math>

Additionally, we can also introduce the following defining relations:
* <math>a[5]2 = a[4]a</math>
* <math>a[5]0 = 1</math>
* <math>a[5](-1) = 0</math>
* <math>a[5](-2) = -1</math>
* <math>a[5](b+1) = a[4](a[5]b)</math>

Other than the trivial cases shown above, pentation generates extremely large numbers very quickly. As a result, there are only a few non-trivial cases that produce numbers that can be written in conventional notation, which are all listed below. 

Some of these numbers are written in [[Power tower (mathematics)|power tower]] notation due to their extreme size. Note that <math> \exp_{10}(n) = 10^n </math>.

* <math>2[5]2 = 2[4]2 = 2^2 = 4</math>
* <math>2[5]3 = 2[4](2[5]2) = 2[4](2[4]2) = 2[4]4 = 2^{2^{2^2}} = 2^{2^4} = 2^{16} = 65,536</math>
* <math>2[5]4 = 2[4](2[5]3) = 2[4](2[4](2[4]2)) = 2[4](2[4]4) = 2[4]65,536 = 2^{2^{2^{\cdot^{\cdot^{\cdot^{2}}}}}} \mbox{ (a power tower of height 65,536) } \approx \exp_{10}^{65,533}(4.29508)</math>.
* <math>2[5]5 = 2[4](2[5]4) = 2[4](2[4](2[4](2[4]2))) = 2[4](2[4](2[4]4)) = 2[4](2[4]65,536) = 2^{2^{2^{\cdot^{\cdot^{\cdot^{2}}}}}} \mbox{ (a power tower of height 2[4]65,536) } \approx \exp_{10}^{2[4]65,536-3}(4.29508)</math>
* <math>3[5]2 = 3[4]3 = 3^{3^3} = 3^{27} = 7,625,597,484,987</math>
* <math>3[5]3 = 3[4](3[5]2) = 3[4](3[4]3) = 3[4]7,625,597,484,987 = 3^{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}} \mbox{ (a power tower of height 7,625,597,484,987) } \approx \exp_{10}^{7,625,597,484,986}(1.09902)</math>
* <math>3[5]4 = 3[4](3[5]3) = 3[4](3[4](3[4]3)) = 3[4](3[4]7,625,597,484,987) = 3^{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}} \mbox{ (a power tower of height 3[4]7,625,597,484,987) } \approx \exp_{10}^{3[4]7,625,597,484,987-1}(1.09902)</math>
* <math>4[5]2 = 4[4]4 =  4^{4^{4^4}} = 4^{4^{256}} \approx \exp_{10}^3(2.19)</math> (a number with over 10<sup>153</sup> digits)
* <math>5[5]2 = 5[4]5 = 5^{5^{5^{5^5}}} = 5^{5^{5^{3125}}} \approx \exp_{10}^4(3.33928)</math> (a number with more than 10<sup>10<sup>2184</sup></sup> digits)

==See also==
*[[Ackermann function]]
*[[Large numbers]]
*[[Graham's number]]
*[[History of large numbers]]

==References==
{{reflist}}

{{Hyperoperations}}
{{Large numbers}}

[[Category:Exponentials]]
[[Category:Large numbers]]
[[Category:Operations on numbers]]