User:TachyonJack/Fun with large numbers

Fun with Large numbers

Using Conways Chained Arrow Notation

$F_{0}(n)\equiv 1$

$F_{1}(n)\equiv n$

$F_{2}(n)\equiv {\begin{matrix}\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\\ \ n{\mbox{ copies of }}n\end{matrix}}$

$F_{3}(n)\equiv \left.{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;\vdots \qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right\}n{\mbox{ layers}}$

$F_{4}(n)\equiv \ \ \ \underbrace {(n{\mbox{ layers}})\left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;\vdots \qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.\left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;{\begin{matrix}\vdots \\\vdots \end{matrix}}\qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.{\Bigg \{}\cdots \cdots \cdots \cdots \cdots \cdots \left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;{\begin{matrix}\vdots \\\vdots \\\vdots \end{matrix}}\qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.} _{\begin{matrix}n{\mbox{ towers}}\end{matrix}}$

$F_{5}(n)\equiv \ \ \left.{\begin{matrix}&\underbrace {n{\mbox{ layers}}\left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;\vdots \qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.\left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;{\begin{matrix}\vdots \\\vdots \end{matrix}}\qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.{\Bigg \{}\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;{\begin{matrix}\vdots \\\vdots \\\vdots \end{matrix}}\qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.} \\&\underbrace {\begin{matrix}\vdots \\\vdots \\\vdots \end{matrix}} \\&\underbrace {n{\mbox{ layers}}\left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;\vdots \qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.\left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;{\begin{matrix}\vdots \\\vdots \end{matrix}}\qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.{\Bigg \{}\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;{\begin{matrix}\vdots \\\vdots \\\vdots \end{matrix}}\qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.} \\&\underbrace {n{\mbox{ layers}}\left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;\vdots \qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.\left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;{\begin{matrix}\vdots \\\vdots \end{matrix}}\qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.{\Bigg \{}\cdots \cdots \cdots \cdots \left\{{\begin{matrix}&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \cdots \cdots \rightarrow n} \\&\underbrace {\qquad \;\;{\begin{matrix}\vdots \\\vdots \\\vdots \end{matrix}}\qquad \;\;} \\&\underbrace {n\rightarrow n\rightarrow \cdots \cdots \rightarrow n} \\&\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\&n{\mbox{ copies of }}n\end{matrix}}\right.} \\&{\begin{matrix}n{\mbox{ towers}}\end{matrix}}\end{matrix}}\right\}n{\mbox{ Super-Layers}}$ $G(n)\equiv F_{n}(n)$