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Concept
Pentation
Summary
In mathematics, pentation (or hyper-5) is the next hyperoperation 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. For instance, using hyperoperation notation for pentation and tetration, means 2 to itself 2 times, or . This can then be reduced to
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.
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. 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.
In Conway chained arrow notation, .
Another proposed notation is , though this is not extensible to higher hyperoperations.
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 .
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