Conway’s chained arrow notation

Conway’s chained arrow notation is a way of writing numbers even larger than those provided by the up arrow notation. We define $m\to n\to p=m^{(p+2)}n=m\underset{p}{\underbrace{\uparrow \mathrm{\cdots }\uparrow }}n$ and $m\to n=m\to n\to 1=m^n$. Longer chains are evaluated by

$$m\to \mathrm{\cdots }\to n\to p\to 1=m\to \mathrm{\cdots }\to n\to p$$

$$m\to \mathrm{\cdots }\to n\to 1\to q=m\to \mathrm{\cdots }\to n$$

and

$$m\to \mathrm{\cdots }\to n\to p+1\to q+1=m\to \mathrm{\cdots }\to n\to (m\to \mathrm{\cdots }\to n\to p\to q+1)\to q$$

For example:

$3\to 3\to 2=$

$3\to (3\to 2\to 2)\to 1=$

$3\to (3\to 2\to 2)=$

$3\to (3\to (3\to 1\to 2)\to 1)=$

$3\to (3\to 3\to 1)=$

$3^{3^3}=$

$3^{27}=7625597484987$

A much larger example is:

$3\to 2\to 4\to 4=$

$3\to 2\to (3\to 2\to 3\to 4)\to 3=$

$3\to 2\to (3\to 2\to (3\to 2\to 2\to 4)\to 3)\to 3=$

$3\to 2\to (3\to 2\to (3\to 2\to (3\to 2\to 1\to 4)\to 3)\to 3)\to 3=$

$3\to 2\to (3\to 2\to (3\to 2\to (3\to 2)\to 3)\to 3)\to 3=$

$3\to 2\to (3\to 2\to (3\to 2\to 9\to 3)\to 3)\to 3$

Clearly this is going to be a very large number. Note that, as large as it is, it is proceeding towards an eventual final evaluation, as evidenced by the fact that the final number in the chain is getting smaller.

Title	Conway’s chained arrow notation
Canonical name	ConwaysChainedArrowNotation
Date of creation	2013-03-22 12:58:46
Last modified on	2013-03-22 12:58:46
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	8
Author	Henry (455)
Entry type	Definition
Classification	msc 00A05
Synonym	chained arrow notation
Synonym	chained arrow
Synonym	chained-arrow
Synonym	chained-arrow notation
Synonym	Conway notation
Related topic	KnuthsUpArrowNotation