Pentation

Inmathematics,pentation(orhyper-5) is the fifthhyperoperation.Pentation is defined to be repeatedtetration,similarly to how tetration is repeatedexponentiation,exponentiation is repeatedmultiplication,and multiplication is repeatedaddition.The concept of "pentation" was named by English mathematicianReuben Goodsteinin 1947, when he came up with the naming scheme for hyperoperations.

The first three values of the expressionx[5]2. The value of 3[5]2 is7625597484987;values for higherx,such as 4[5]2, which is about 2.361 × 10^{8.072 × 10¹⁵³}are much too large to appear on the graph.

The numberapentated to the numberbis defined asatetrated to itselfb - 1times. This may variously be denoted as $a[5]b$ , $a\uparrow \uparrow \uparrow b$ , $a\uparrow ^{3}b$ , $a\to b\to 3$ ,or ${_{b}a}$ ,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 $2^{2}=4$ .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 $2^{2^{2^{2}}}=65536$ .

Based on this definition, pentation is only defined whenaandbare bothpositive integers.

Definition

Pentation is the nexthyperoperation(infinitesequenceof arithmetic operations, based on the previous one each time) aftertetrationand before hexation. It is defined asiterated(repeated) tetration (assuming right-associativity). This is similar to as tetration is iterated right-associativeexponentiation.^[1]It is abinary operationdefined with two numbersaandb,whereais tetrated to itselfb − 1times.

The type of hyperoperation is typically denoted by a number in brackets, []. For instance, usinghyperoperationnotation for pentation and tetration, $2[5]3$ means tetrating 2 to itself 2 times, or $2[4](2[4]2)$ .This can then be reduced to $2[4](2^{2})=2[4]4=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65,536.$

Etymology

The word "pentation" was coined byReuben Goodsteinin 1947 from the rootspenta-(five) anditeration.It is part of his general naming scheme forhyperoperations.^[2]

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 ahyperoperationas $a[5]b$ .In this format, $a[3]b$ may be interpreted as the result ofrepeatedly applyingthe function $x\mapsto a[2]x$ ,for $b$ repetitions, starting from the number 1. Analogously, $a[4]b$ ,tetration, represents the value obtained by repeatedly applying the function $x\mapsto a[3]x$ ,for $b$ repetitions, starting from the number 1, and the pentation $a[5]b$ represents the value obtained by repeatedly applying the function $x\mapsto a[4]x$ ,for $b$ repetitions, starting from the number 1.^[3]^[4]This will be the notation used in the rest of the article.

InKnuth's up-arrow notation, $a[5]b$ is represented as $a\uparrow \uparrow \uparrow b$ or $a\uparrow ^{3}b$ .In this notation, $a\uparrow b$ represents the exponentiation function $a^{b}$ and $a\uparrow \uparrow b$ represents tetration. The operation can be easily adapted for hexation by adding another arrow.
InConway chained arrow notation, $a[5]b=a\rightarrow b\rightarrow 3$ .^[5]
Another proposed notation is ${_{b}a}$ ,though this is not extensible to higher hyperoperations.^[6]

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 theAckermann function:if $A(n,m)$ is defined by the Ackermann recurrence $A(m-1,A(m,n-1))$ with the initial conditions $A(1,n)=an$ and $A(m,1)=a$ ,then $a[5]b=A(4,b)$ .^[7]

As tetration, its base operation, has not been extended to non-integer heights, pentation $a[5]b$ is currently only defined for integer values ofaandbwherea> 0 andb≥ −2, and a few other integer values whichmaybe uniquely defined. As with all hyperoperations of order 3 (exponentiation) and higher, pentation has the followingtrivialcases (identities) which holds for all values ofaandbwithin its domain:

$1[5]b=1$
$a[5]1=a$

Additionally, we can also introduce the following defining relations:

$a[5]2=a[4]a$
$a[5]0=1$
$a[5](-1)=0$
$a[5](-2)=-1$
$a[5](b+1)=a[4](a[5]b)$

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 inpower towernotation due to their extreme size. Note that $\exp _{10}(n)=10^{n}$ .

$2[5]2=2[4]2=2^{2}=4$
$2[5]3=2[4](2[5]2)=2[4](2[4]2)=2[4]4=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65,536$
$2[5]4=2[4](2[5]3)=2[4](2[4](2[4]2))=2[4](2[4]4)=2[4]65,536=2^{2^{2^{\cdot ^{\cdot ^{\cdot ^{2}}}}}}{\mbox{ (a power tower of height 65,536) }}\approx \exp _{10}^{65,533}(4.29508)$ .
$2[5]5=2[4](2[5]4)=2[4](2[4](2[4](2[4]2)))=2[4](2[4](2[4]4))=2[4](2[4]65,536)=2^{2^{2^{\cdot ^{\cdot ^{\cdot ^{2}}}}}}{\mbox{ (a power tower of height 2[4]65,536) }}\approx \exp _{10}^{2[4]65,536-3}(4.29508)$
$3[5]2=3[4]3=3^{3^{3}}=3^{27}=7,625,597,484,987$
$3[5]3=3[4](3[5]2)=3[4](3[4]3)=3[4]7,625,597,484,987=3^{3^{3^{\cdot ^{\cdot ^{\cdot ^{3}}}}}}{\mbox{ (a power tower of height 7,625,597,484,987) }}\approx \exp _{10}^{7,625,597,484,986}(1.09902)$
$3[5]4=3[4](3[5]3)=3[4](3[4](3[4]3))=3[4](3[4]7,625,597,484,987)=3^{3^{3^{\cdot ^{\cdot ^{\cdot ^{3}}}}}}{\mbox{ (a power tower of height 3[4]7,625,597,484,987) }}\approx \exp _{10}^{3[4]7,625,597,484,987-1}(1.09902)$
$4[5]2=4[4]4=4^{4^{4^{4}}}=4^{4^{256}}\approx \exp _{10}^{3}(2.19)$ (a number with over 10¹⁵³digits)
$5[5]2=5[4]5=5^{5^{5^{5^{5}}}}=5^{5^{5^{3125}}}\approx \exp _{10}^{4}(3.33928)$ (a number with more than 10^10²¹⁸⁴digits)