Pentation

In mathematics, pentation is the operation of repeated tetration, just as tetration is the operation of repeated exponentiation.

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.

Pentation can be written as a hyperoperation as ${\displaystyle a[5]b}$, or using Knuth's up-arrow notation as ${\displaystyle a\uparrow \uparrow \uparrow b}$ or ${\displaystyle a\uparrow ^{3}b}$. In this notation, ${\displaystyle a\uparrow b}$ represents the exponentiation function ${\displaystyle a^{b}}$, which may be interpreted as the result of repeatedly applying the function ${\displaystyle x\mapsto ax}$, for ${\displaystyle b}$ repetitions, starting from the number 1. Analogously, ${\displaystyle a\uparrow \uparrow b}$, tetration, represents the value obtained by repeatedly applying the function ${\displaystyle x\mapsto a\uparrow x}$, for ${\displaystyle b}$ repetitions, starting from the number 1. And the pentation ${\displaystyle a\uparrow \uparrow \uparrow b}$ represents the value obtained by repeatedly applying the function ${\displaystyle x\mapsto a\uparrow \uparrow x}$, for ${\displaystyle b}$ repetitions, starting from the number 1. Alternatively, in Conway chained arrow notation, ${\displaystyle a\uparrow \uparrow \uparrow b=a\rightarrow b\rightarrow 3}$. Another proposed notation is ${\displaystyle {_{b}a}}$, though this is not extensible to higher hyperoperations.

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