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## 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. 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, $23$ means tetrating 2 to itself 2 times, or $2(22)$ . This can then be reduced to $2(2^{2})=24=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65536.$  The first three values of the expression x2. The value of 32 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.

## 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 $ab$ . In this format, $ab$  may be interpreted as the result of repeatedly applying the function $x\mapsto ax$ , for $b$  repetitions, starting from the number 1. Analogously, $ab$ , tetration, represents the value obtained by repeatedly applying the function $x\mapsto ax$ , for $b$  repetitions, starting from the number 1, and the pentation $ab$  represents the value obtained by repeatedly applying the function $x\mapsto ax$ , for $b$  repetitions, starting from the number 1. This will be the notation used in the rest of the article.
• In Knuth's up-arrow notation, $ab$  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.
• In Conway chained arrow notation, $ab=a\rightarrow b\rightarrow 3$ .
• Another proposed notation is ${_{b}a}$ , though this is not extensible to higher hyperoperations.

## 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 $A(n,m)$  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 $ab=A(4,b)$ .

As tetration, its base operation, has not been extended to non-integer heights, pentation $ab$  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:

• $1b=1$
• $a1=a$

• $a0=1$
• $a(-1)=0$

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:

• $22=22=2^{2}=4$
• $23=2(22)=24=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65,536$
• $24=2(2(22))=2(24)=265536=2^{2^{2^{\cdot ^{\cdot ^{\cdot ^{2}}}}}}{\mbox{ (a power tower of height 65,536) }}\approx \exp _{10}^{65,533}(4.29508)$  (shown here in iterated exponential notation as it is far too large to be written in conventional notation. Note $\exp _{10}(n)=10^{n}$ )
• $32=33=3^{3^{3}}=3^{27}=7,625,597,484,987$
• $33=3(33)=37,625,597,484,987={\underset {{\text{3 is repeated 33 times}}=3^{3^{3}}}{3^{3^{.^{.^{.^{3}}}}}}}{\mbox{ (a power tower of height 7,625,597,484,987) }}\approx \exp _{10}^{7,625,597,484,986}(1.09902)$
• $42=44=4^{4^{4^{4}}}=4^{4^{256}}\approx \exp _{10}^{3}(2.19)$  (a number with over 10153 digits)
• $52=55=5^{5^{5^{5^{5}}}}=5^{5^{5^{3125}}}\approx \exp _{10}^{4}(3.33928)$  (a number with more than 10102184 digits)