# 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 $m\rightarrow n\rightarrow p=m^{(p+2)}n=m\underbrace{\uparrow\cdots\uparrow}_{p}n$ and $m\rightarrow n=m\rightarrow n\rightarrow 1=m^{n}$. Longer chains are evaluated by

 $m\rightarrow\cdots\rightarrow n\rightarrow p\rightarrow 1=m\rightarrow\cdots% \rightarrow n\rightarrow p$
 $m\rightarrow\cdots\rightarrow n\rightarrow 1\rightarrow q=m\rightarrow\cdots\rightarrow n$

and

 $m\rightarrow\cdots\rightarrow n\rightarrow p+1\rightarrow q+1=m\rightarrow% \cdots\rightarrow n\rightarrow(m\rightarrow\cdots\rightarrow n\rightarrow p% \rightarrow q+1)\rightarrow q$

For example:

 $\displaystyle 3\rightarrow 3\rightarrow 2=$ $\displaystyle 3\rightarrow(3\rightarrow 2\rightarrow 2)\rightarrow 1=$ $\displaystyle 3\rightarrow(3\rightarrow 2\rightarrow 2)=$ $\displaystyle 3\rightarrow(3\rightarrow(3\rightarrow 1\rightarrow 2)% \rightarrow 1)=$ $\displaystyle 3\rightarrow(3\rightarrow 3\rightarrow 1)=$ $\displaystyle 3^{3^{3}}=$ $\displaystyle 3^{27}=7625597484987$

A much larger example is:

 $\displaystyle 3\rightarrow 2\rightarrow 4\rightarrow 4=$ $\displaystyle 3\rightarrow 2\rightarrow(3\rightarrow 2\rightarrow 3\rightarrow 4% )\rightarrow 3=$ $\displaystyle 3\rightarrow 2\rightarrow(3\rightarrow 2\rightarrow(3\rightarrow 2% \rightarrow 2\rightarrow 4)\rightarrow 3)\rightarrow 3=$ $\displaystyle 3\rightarrow 2\rightarrow(3\rightarrow 2\rightarrow(3\rightarrow 2% \rightarrow(3\rightarrow 2\rightarrow 1\rightarrow 4)\rightarrow 3)\rightarrow 3% )\rightarrow 3=$ $\displaystyle 3\rightarrow 2\rightarrow(3\rightarrow 2\rightarrow(3\rightarrow 2% \rightarrow(3\rightarrow 2)\rightarrow 3)\rightarrow 3)\rightarrow 3=$ $\displaystyle 3\rightarrow 2\rightarrow(3\rightarrow 2\rightarrow(3\rightarrow 2% \rightarrow 9\rightarrow 3)\rightarrow 3)\rightarrow 3$

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