Pentation

Short description: Arithmetic operation

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. 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, $2[5]3$ 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 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 $a[5]b$. In this format, $a[3]b$ may be interpreted as the result of repeatedly applying the 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. This will be the notation used in the rest of the article.

• In Knuth'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.
• In Conway chained arrow notation, $a[5]b=a\to b\to 3$.
• 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 $a[5]b=A(4,b)$.

As tetration, its base operation, has not been extended to non-integer heights, pentation $a[5]b$ 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:

• $1[5]b=1$
• $a[5]1=a$

Additionally, we can also define:

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

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

• $2[5]2=2[4]2=2^2=4$
• $2[5]3=2[4](2[5]2)=2[4](2[4]2)=2[4]4=2^{2^{2^2}}=2^{2^4}=2^{16}=65,536$
• $2[5]4=2[4](2[5]3)=2[4](2[4](2[4]2))=2[4](2[4]4)=2[4]65,536=2^{2^{2^{{\cdot }^{{\cdot }^{{\cdot }^2}}}}}\text{ (a power tower of height 65,536) }\approx {\exp }_{10}^{65,533}(4.29508)$ (shown here in iterated exponential notation as it is far too large to be written in conventional notation. Note ${\exp }_{10}(n)=10^n$)
• $2[5]5=2[4](2[5]4)=2[4](2[4](2[4](2[4]2)))=2[4](2[4](2[4]4))=2[4](2[4]65,536)=2^{2^{2^{{\cdot }^{{\cdot }^{{\cdot }^2}}}}}\text{ (a power tower of height 2[4]65,536) }\approx {\exp }_{10}^{2[4]65,536-3}(4.29508)$
• $3[5]2=3[4]3=3^{3^3}=3^{27}=7,625,597,484,987$
• $3[5]3=3[4](3[5]2)=3[4](3[4]3)=3[4]7,625,597,484,987=3^{3^{3^{{\cdot }^{{\cdot }^{{\cdot }^3}}}}}\text{ (a power tower of height 7,625,597,484,987) }\approx {\exp }_{10}^{7,625,597,484,986}(1.09902)$
• $3[5]4=3[4](3[5]3)=3[4](3[4](3[4]3))=3[4](3[4]7,625,597,484,987)=3^{3^{3^{{\cdot }^{{\cdot }^{{\cdot }^3}}}}}\text{ (a power tower of height 3[4]7,625,597,484,987) }\approx {\exp }_{10}^{3[4]7,625,597,484,987-1}(1.09902)$
• $4[5]2=4[4]4=4^{4^{4^4}}=4^{4^{256}}\approx {\exp }_{10}^3(2.19)$ (a number with over 10¹⁵³ digits)
• $5[5]2=5[4]5=5^{5^{5^{5^5}}}=5^{5^{5^{3125}}}\approx {\exp }_{10}^4(3.33928)$ (a number with more than 10¹⁰²¹⁸⁴ digits)

See also

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