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The wikipage for Knuth's up-arrow notation says $a \uparrow ^n b$ is defined recursively for integer $a$ and non-negative integers $b$ and $n$ like so: \begin{equation} a\uparrow^n b = \begin{cases} a^b, & \text{if }n=1; \\ 1, & \text{if }n\ge 1\text{ and }b=0; \\ a\uparrow^{n-1}(a\uparrow^{n}(b-1)), & \text{otherwise } \end{cases} \end{equation} However, some negative values of $a$ will push the $b$ in further iterations into the negatives, and consequently into infinitely small $b$ - with no defined value for the expression. Is there a different definition? Is it not supposed to be defined for all integer $a$ and non-negative integers $b$ and $n$?

Example:

$-1 \uparrow^3 2 = -1 \uparrow^2 (-1 \uparrow^3 1) = -1 \uparrow^2 (-1 \uparrow^2 (-1 \uparrow^3 0)) = -1 \uparrow^2 (-1 \uparrow^2 1) = -1 \uparrow^2 -1$

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The main purpose of this operation is to create insanely large numbers , already $$3\uparrow \uparrow \uparrow 3$$ is huge ( a power tower of $3^{27}$ threes).

So, usually the arguments are positive integers. I am not sure whether a generalization to negative or even real numbers actually makes sense, but apparently some tries have been made.

The tetration is interesting also for real numbers, for example the infinite power tower $$x^{x^{x^...}}$$ but the operations after that are difficult to extend.

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  • $\begingroup$ So you reckon the source should say "non-negative (integer?) $a$" rather than "integer $a$"? $\endgroup$ – Athere Aug 1 '19 at 13:33
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    $\begingroup$ To my opinion, this would have been better. $\endgroup$ – Peter Aug 1 '19 at 16:35

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