Knuth's up arrow notation

Knuth's up arrow noation is a way of writing numbers which would be unwieldy in standard decimal notation. It expands on the exponential notation $m\uparrow n=m^n$ . Define $m\uparrow\uparrow 0=1$ and $m \uparrow\uparrow n=m\uparrow(m\uparrow\uparrow [n-1])$ .

Obviously $m\uparrow\uparrow 1=m^1=m$ , so $3\uparrow\uparrow 2=3^{3\uparrow\uparrow 1}=3^3=27$ , but $2\uparrow\uparrow 3=2^{2 \uparrow\uparrow 2}=2^{2^{2\uparrow\uparrow 1}}=2^{(2^2)}=16$ .

In general, $m\uparrow\uparrow n=m^{m^{\cdots^m}}$ , a tower of height $n$ .

Clearly, this process can be extended: $m\uparrow\uparrow\uparrow 0=1$ and $m\uparrow\uparrow\uparrow n=m\uparrow\uparrow(m\uparrow\uparrow\uparrow [n-1])$ .

An alternate notation is to write $m^{(i)}n$ for $m\underbrace{\uparrow\cdots\uparrow}_{i-2 {~times}}n$ . ($i-2$ times because then $m^{(2)}n=m\cdot n$ and $m^{(1)}n=m+n$ .) Then in general we can define $m^{(i)}n=m^{(i-1)}(m^{(i)}(n-1))$ .

To get a sense of how quickly these numbers grow, $3\uparrow\uparrow\uparrow 2=3\uparrow\uparrow 3$ is more than seven and a half trillion, and the numbers continue to grow much more than exponentially.