# Domain for Knuth's up-arrow

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: $$$$a\uparrow^n b = \begin{cases} a^b, & \text{if }n=1; \\ 1, & \text{if }n\ge 1\text{ and }b=0; \\ a\uparrow^{n-1}(a\uparrow^{n}(b-1)), & \text{otherwise } \end{cases}$$$$ However, some negative values of $$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:

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

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).
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.
• So you reckon the source should say "non-negative (integer?) $a$" rather than "integer $a$"? – Athere Aug 1 '19 at 13:33