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, $m\uparrow \uparrow n=m^{m^{{\mathrm{\cdots }}^m}}$, 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 $m\underset{i-2\text{ times}}{\underbrace{\uparrow \mathrm{\cdots }\uparrow }}n$. ($i-2$ 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.

Title	Knuth’s up arrow notation
Canonical name	KnuthsUpArrowNotation
Date of creation	2013-03-22 12:58:43
Last modified on	2013-03-22 12:58:43
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	7
Author	Henry (455)
Entry type	Definition
Classification	msc 00A05
Synonym	up-arrow
Synonym	up arrow
Synonym	up-arrow notation
Synonym	up arrow notation
Synonym	Knuth notation
Related topic	ConwaysChainedArrowNotation