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Pentation

## Summary

In mathematics, pentation (or hyper-5) is the next hyperoperation after tetration and before hexation. It is defined as iterated (repeated) tetration, just as tetration is iterated exponentiation.[1] It is a binary operation defined with two numbers a and b, where a is tetrated to itself b-1 times. For instance, using hyperoperation notation for pentation and tetration, ${\displaystyle 2[5]3}$ means tetrating 2 to itself 2 times, or ${\displaystyle 2[4](2[4]2)}$. This can then be reduced to ${\displaystyle 2[4](2^{2})=2[4]4=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65536.}$

The first three values of the expression x[5]2. The value of 3[5]2 is about 7.626 × 1012; values for higher x are much too large to appear on the graph.

## Etymology

The word "pentation" was coined by Reuben Goodstein in 1947 from the roots penta- (five) and iteration. It is part of his general naming scheme for hyperoperations.[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 a hyperoperation as ${\displaystyle a[5]b}$ . In this format, ${\displaystyle a[3]b}$  may be interpreted as the result of repeatedly applying the function ${\displaystyle x\mapsto a[2]x}$ , for ${\displaystyle b}$  repetitions, starting from the number 1. Analogously, ${\displaystyle a[4]b}$ , tetration, represents the value obtained by repeatedly applying the function ${\displaystyle x\mapsto a[3]x}$ , for ${\displaystyle b}$  repetitions, starting from the number 1, and the pentation ${\displaystyle a[5]b}$  represents the value obtained by repeatedly applying the function ${\displaystyle x\mapsto a[4]x}$ , for ${\displaystyle b}$  repetitions, starting from the number 1.[3][4] This will be the notation used in the rest of the article.
• In Knuth's up-arrow notation, ${\displaystyle a[5]b}$  is represented as ${\displaystyle a\uparrow \uparrow \uparrow b}$  or ${\displaystyle a\uparrow ^{3}b}$ . In this notation, ${\displaystyle a\uparrow b}$  represents the exponentiation function ${\displaystyle a^{b}}$  and ${\displaystyle a\uparrow \uparrow b}$  represents tetration. The operation can be easily adapted for hexation by adding another arrow.
• Another proposed notation is ${\displaystyle {_{b}a}}$ , 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 the Ackermann function: if ${\displaystyle A(n,m)}$  is defined by the Ackermann recurrence ${\displaystyle A(m-1,A(m,n-1))}$  with the initial conditions ${\displaystyle A(1,n)=an}$  and ${\displaystyle A(m,1)=a}$ , then ${\displaystyle a[5]b=A(4,b)}$ .[7]

As tetration, its base operation, has not been extended to non-integer heights, pentation ${\displaystyle a[5]b}$  is currently only defined for integer values of a and b where a > 0 and b ≥ −1, 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 trivial cases (identities) which holds for all values of a and b within its domain:

• ${\displaystyle 1[5]b=1}$
• ${\displaystyle a[5]1=a}$

• ${\displaystyle a[5]0=1}$
• ${\displaystyle a[5](-1)=0}$

Other than the trivial cases shown above, pentation generates extremely large numbers very quickly such that there are only a few non-trivial cases that produce numbers that can be written in conventional notation, as illustrated below:

• ${\displaystyle 2[5]2=2[4]2=2^{2}=4}$
• ${\displaystyle 2[5]3=2[4](2[4]2)=2[4]4=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65,536}$
• ${\displaystyle 2[5]4=2[4](2[4](2[4]2))=2[4](2[4]4)=2[4]65536=2^{2^{2^{\cdot ^{\cdot ^{\cdot ^{2}}}}}}{\mbox{ (a power tower of height 65,536) }}\approx \exp _{10}^{65,533}(4.29508)}$  (shown here in iterated exponential notation as it is far too large to be written in conventional notation. Note ${\displaystyle \exp _{10}(n)=10^{n}}$ )
• ${\displaystyle 3[5]2=3[4]3=3^{3^{3}}=3^{27}=7,625,597,484,987}$
• ${\displaystyle 3[5]3=3[4](3[4]3)=3[4]7,625,597,484,987={\underset {{\text{3 is repeated 3[4]3 times}}=3^{3^{3}}}{3^{3^{.^{.^{.^{3}}}}}}}{\mbox{ (a power tower of height 7,625,597,484,987) }}\approx \exp _{10}^{7,625,597,484,986}(1.09902)}$
• ${\displaystyle 4[5]2=4[4]4=4^{4^{4^{4}}}=4^{4^{256}}\approx \exp _{10}^{3}(2.19)}$  (a number with over 10153 digits)
• ${\displaystyle 5[5]2=5[4]5=5^{5^{5^{5^{5}}}}=5^{5^{5^{3125}}}\approx \exp _{10}^{4}(3.33928)}$  (a number with more than 10102184 digits)