Book Hotels in Pentation
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.
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 .
It is not known how to extend pentation to complex or non-integer values.
Using super-logarithms, can be done when b is negative or 0, provided it is not too negative. For all positive integer values of a negative pentation is as follows:
Any other case of this type of pentation produces an undefined result, since integer tetration does not take on the value -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 , is undefined because tetration is only defined for b greater than -2, while if a is zero, we obtain the presumably-indeterminate form .
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: