Pentation

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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, [math]\displaystyle{ 2[5]3 }[/math] means tetrating 2 to itself 2 times, or [math]\displaystyle{ 2[4](2[4]2) }[/math]. This can then be reduced to [math]\displaystyle{ 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 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 [math]\displaystyle{ a[5]b }[/math]. In this format, [math]\displaystyle{ a[3]b }[/math] may be interpreted as the result of repeatedly applying the function [math]\displaystyle{ x\mapsto a[2]x }[/math], for [math]\displaystyle{ b }[/math] repetitions, starting from the number 1. Analogously, [math]\displaystyle{ a[4]b }[/math], tetration, represents the value obtained by repeatedly applying the function [math]\displaystyle{ x\mapsto a[3]x }[/math], for [math]\displaystyle{ b }[/math] repetitions, starting from the number 1, and the pentation [math]\displaystyle{ a[5]b }[/math] represents the value obtained by repeatedly applying the function [math]\displaystyle{ x\mapsto a[4]x }[/math], for [math]\displaystyle{ b }[/math] 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, [math]\displaystyle{ a[5]b }[/math] is represented as [math]\displaystyle{ a \uparrow \uparrow \uparrow b }[/math] or [math]\displaystyle{ a \uparrow^{3}b }[/math]. In this notation, [math]\displaystyle{ a\uparrow b }[/math] represents the exponentiation function [math]\displaystyle{ a^b }[/math] and [math]\displaystyle{ 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]\displaystyle{ a[5]b = a\rightarrow b\rightarrow 3 }[/math].^[5]
• Another proposed notation is [math]\displaystyle{ {_{b}a} }[/math], 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 [math]\displaystyle{ A(n,m) }[/math] is defined by the Ackermann recurrence [math]\displaystyle{ A(m-1,A(m,n-1)) }[/math] with the initial conditions [math]\displaystyle{ A(1,n)=an }[/math] and [math]\displaystyle{ A(m,1)=a }[/math], then [math]\displaystyle{ a[5]b=A(4,b) }[/math].^[7]

As tetration, its base operation, has not been extended to non-integer heights, pentation [math]\displaystyle{ 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 trivial cases (identities) which holds for all values of a and b within its domain:

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

Additionally, we can also define:

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

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:

• [math]\displaystyle{ 2[5]2 = 2[4]2 = 2^2 = 4 }[/math]
• [math]\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 }[/math]
• [math]\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) }[/math] (shown here in iterated exponential notation as it is far too large to be written in conventional notation. Note [math]\displaystyle{ \exp_{10}(n) = 10^n }[/math])
• [math]\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) }[/math]
• [math]\displaystyle{ 3[5]2 = 3[4]3 = 3^{3^3} = 3^{27} = 7,625,597,484,987 }[/math]
• [math]\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) }[/math]
• [math]\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) }[/math]
• [math]\displaystyle{ 4[5]2 = 4[4]4 = 4^{4^{4^4}} = 4^{4^{256}} \approx \exp_{10}^3(2.19) }[/math] (a number with over 10¹⁵³ digits)
• [math]\displaystyle{ 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 10¹⁰²¹⁸⁴ digits)

See also

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

References

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