Book Hotels in Pentation
Population: Unkown
Latitude: 19.846716
Longitude: 39.532588
Source: WikiPedia
Description:
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 realworld applications. It is noncommutative, and therefore has two inverse functions, which might be named the pentaroot and the pentalogarithm (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 uparrow notation as or .
It is not known how to extend pentation to complex or noninteger values.[citation needed]
Using superlogarithms, 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 presumablyindeterminate form .
As its base operation (tetration) has not been extended to noninteger 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 nontrivial cases that produce numbers that can be written in conventional notation, as illustrated below:


