Pentation
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Pentation is the operation of repeated tetration, just as tetration is the operation of repeated exponentation and is a hyperoperation. Like tetration, it has little real-world applications. It is non-commutative, and therefore has two inverse functions, which might be named the penta-root and the penta-logarithm (analogous to the two inverse functions for exponentiation: nth root function and logarithm). Pentation also bounds the elementary recursive functions[citation needed].
The word "Pentation" was coined by Reuben Goodstein from the roots penta- (five) and iteration). It is part of his general naming scheme for hyperoperations.
Pentation can be written in Knuth's up-arrow notation as or .
Extension
It is not known how to extend pentation to complex or non-integer values.[citation needed]
Extension to zero and negative numbers
Using super-logarithms, can be done when b is negative or 0, but doing so is a lot more limited. For all positive integer values of a negative pentation is as follows:
- if a > 1.
- if a > 1.
- if a > 1.
It can also be done when a is negative, but this is only the case when a is equal to -1. For all positive integer values of b, the three possible answers that you can get for are shown below:
- if b is congruent to 1 modulo 3.
- if b is congruent to 2 modulo 3.
- if b is congruent to 0 modulo 3.
Any other case of this type of pentation produces an undefined result.
Selected values
As its base operation (tetration) 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 ≥ 0, and a few other integer values which may be uniquely defined. Like all other 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:
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:
- (shown here in iterated exponential notation as it's far too large to be written in conventional notation. Note )
- (a number with over 10153 digits)
- (a number with more than 10102184 digits)
References
- Tetration Forum by Jay D. Fox