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, such as 4[5]2, which is about 2.361 × 10^{8.072 × 10153} are much too large to appear on the graph.

In mathematics, pentation (or hyper-5) is the next hyperoperation (infinite sequence of arithmetic operations) after tetration and before hexation. It is defined as iterated (repeated) tetration (assuming right-associativity), just as tetration is iterated right-associative exponentiation.^[1] It is a binary operation defined with two numbers a and b, where a is tetrated to itself b − 1 times. (The number in the brackets, [], represents the type of hyperoperation.) 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 ${\displaystyle 2[4](2^{2})=2[4]4=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65,536.}$

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.
• In Conway chained arrow notation, ${\displaystyle a[5]b=a\rightarrow b\rightarrow 3}$.^[5]
• 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 ≥ −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 trivial cases (identities) which holds for all values of a and b within its domain:

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

Additionally, we can also define:

• ${\displaystyle a[5]2=a[4]a}$
• ${\displaystyle a[5]0=1}$
• ${\displaystyle a[5](-1)=0}$
• ${\displaystyle a[5](-2)=-1}$
• ${\displaystyle a[5](b+1)=a[4](a[5]b)}$

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:

• ${\displaystyle 2[5]2=2[4]2=2^{2}=4}$
• ${\displaystyle 2[5]3=2[4](2[5]2)=2[4](2[4]2)=2[4]4=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65,536}$
• ${\displaystyle 2[5]4=2[4](2[5]3)=2[4](2[4](2[4]2))=2[4](2[4]4)=2[4]65,536=2^{2^{2^{\cdot ^{\cdot ^{\cdot ^{2}}}}}}{\mbox{ (a power tower of height 65,536) }}\approx \exp _{10}^{65,533}(4.29508)}$ (shown here in iterated exponential notation as it is far too large to be written in conventional notation. Note ${\displaystyle \exp _{10}(n)=10^{n}}$)
• ${\displaystyle 2[5]5=2[4](2[5]4)=2[4](2[4](2[4](2[4]2)))=2[4](2[4](2[4]4))=2[4](2[4]65,536)=2^{2^{2^{\cdot ^{\cdot ^{\cdot ^{2}}}}}}{\mbox{ (a power tower of height 2[4]65,536) }}\approx \exp _{10}^{2[4]65,536-3}(4.29508)}$
• ${\displaystyle 3[5]2=3[4]3=3^{3^{3}}=3^{27}=7,625,597,484,987}$
• ${\displaystyle 3[5]3=3[4](3[5]2)=3[4](3[4]3)=3[4]7,625,597,484,987=3^{3^{3^{\cdot ^{\cdot ^{\cdot ^{3}}}}}}{\mbox{ (a power tower of height 7,625,597,484,987) }}\approx \exp _{10}^{7,625,597,484,986}(1.09902)}$
• ${\displaystyle 3[5]4=3[4](3[5]3)=3[4](3[4](3[4]3))=3[4](3[4]7,625,597,484,987)=3^{3^{3^{\cdot ^{\cdot ^{\cdot ^{3}}}}}}{\mbox{ (a power tower of height 3[4]7,625,597,484,987) }}\approx \exp _{10}^{3[4]7,625,597,484,987-1}(1.09902)}$
• ${\displaystyle 4[5]2=4[4]4=4^{4^{4^{4}}}=4^{4^{256}}\approx \exp _{10}^{3}(2.19)}$ (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)

See also

• Ackermann function
• Large numbers
• Graham's number
• History of large numbers

References