Editing Conway chained arrow notation (section)

== Extension by Peter Hurford ==
Peter Hurford, a web developer and statistician, has defined an extension to this notation:

<math>a \rightarrow_b c = \underbrace{a\rightarrow_{b-1} a\rightarrow_{b-1} a\rightarrow_{b-1} \dots \rightarrow_{b-1} a\rightarrow_{b-1} a\rightarrow_{b-1} a}_{c \text{ arrows}}</math>

<math>a \rightarrow_1 b = a \rightarrow b</math>

All normal rules are unchanged otherwise.

<math>a \rightarrow_2 (a-1)</math> is already equal to the aforementioned <math>cg(a)</math>, and the function <math>f(n) = n \rightarrow_n n</math> is much faster growing than Conway and Guy's <math>cg(n)</math>.

Note that expressions like <math>a \rightarrow_b c \rightarrow_d e</math> are illegal if <math>b</math> and <math>d</math> are different numbers; a chain must have only one type of right-arrow.

However, if we modify this slightly such that:

<math>a \rightarrow_b c \rightarrow_d e = a \rightarrow_b \underbrace{c \rightarrow_{d-1} c \rightarrow_{d-1} c \rightarrow_{d-1} \dots \rightarrow_{d-1} c \rightarrow_{d-1} c \rightarrow_{d-1} c}_{e \text{ arrows}}</math>

then not only does <math>a \rightarrow_b c \rightarrow_d e</math> become legal, but the notation as a whole becomes much stronger.<ref>{{Cite news|url=http://www.greatplay.net/essays/large-numbers-part-ii-graham-and-conway|archive-url=https://archive.today/20130625000516/http://www.greatplay.net/essays/large-numbers-part-ii-graham-and-conway|url-status=dead|archive-date=2013-06-25|title=Large Numbers, Part 2: Graham and Conway - Greatplay.net|date=2013-06-25|work=archive.is|access-date=2018-02-18}}</ref>