In
mathematics, pentation (or hyper-5) is the fifth
hyperoperation. Pentation is defined to be repeated
tetration, similarly to how tetration is repeated
exponentiation, exponentiation is repeated
multiplication, and multiplication is repeated
addition. The concept of "pentation" was named by English mathematician
Reuben Goodstein in 1947, when he came up with the naming scheme for hyperoperations.
The number a pentated to the number b is defined as a tetrated to itself b - 1 times. This may variously be denoted as , , , , or , depending on one's choice of notation.
For example, 2 pentated to the 2 is 2 tetrated to the 2, or 2 raised to the power of 2, which is . As another example, 2 pentated to the 3 is 2 tetrated to the result of 2 tetrated to the 2. Since 2 tetrated to the 2 is 4, 2 pentated to the 3 is 2 tetrated to the 4, which is .
Based on this definition, pentation is only defined when a and b are both
positive integers.
Definition
Pentation is the next
hyperoperation (infinite
sequence of arithmetic operations) after
tetration and before
hexation. It is defined as
iterated (repeated) tetration (assuming right-associativity). This is similar to 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 type of hyperoperation is typically denoted by a number in brackets, []. For instance, using
hyperoperation notation for pentation and tetration, means tetrating 2 to itself 3 times, or . This can then be reduced to
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 . In this format, may be interpreted as the result of
repeatedly applying the function , for repetitions, starting from the number 1. Analogously, , tetration, represents the value obtained by repeatedly applying the function , for repetitions, starting from the number 1, and the pentation represents the value obtained by repeatedly applying the function , for 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, is represented as or . In this notation, represents the exponentiation function and represents tetration. The operation can be easily adapted for hexation by adding another arrow.
Another proposed notation is , 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 is defined by the Ackermann recurrence with the initial conditions and , then .[7]
As tetration, its base operation, has not been extended to non-integer heights, pentation 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:
Additionally, we can also introduce the following defining relations:
Other than the trivial cases shown above, pentation generates extremely large numbers very quickly. As a result, there are only a few non-trivial cases that produce numbers that can be written in conventional notation, which are all listed below.
Some of these numbers are written in
power tower notation due to their extreme size. Note that .