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Hexation is the operation of repeated pentation, just as pentation is the operation of repeated tetration and is a hyperoperation. Like pentation and tetration, it has little real-world applications, and has few applications even in mathematics. It is non-commutative, and therefore has two inverse functions, which might be named the hexa-root and the hexa-logarithm (analogous to the two inverse functions for exponentiation: nth root function and logarithm). Hexation is not elementary recursive.

The word "Hexation" was coined by Reuben Goodstein from the roots hexa- (six) and iteration. It is part of his general naming scheme for hyperoperations.

Hexation can be written in Knuth's up-arrow notation as ${\displaystyle a\uparrow \uparrow \uparrow \uparrow b}$ or ${\displaystyle a\uparrow ^{4}b}$.

Extension

It is not known how to extend hexation to complex or non-integer values.

Selected values

As its base operation (pentation) has not been extended to non-integer heights, hexation ${\displaystyle a\uparrow ^{4}b}$ is currently only defined for integer values of a and b where a > 0 and b ≥ 0. 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:

• ${\displaystyle 1\uparrow ^{4}b=1}$
• ${\displaystyle a\uparrow ^{4}1=a}$

Other than the trivial cases shown above, hexation generates extremely large numbers very quickly such that there is only one fnon-trivial case that produces numbers that can be written in conventional notation, as illustrated below:

• ${\displaystyle 2\uparrow ^{4}2=4}$

This is the reason why hexation is not normally used in mathematics: the numbers it produces are so vast that they have little use.