グラハム数
F := F 7 ( 12 ) = F ( F ( F ( F ( F ( F ( F ( 12 ) ) ) ) ) ) ) = 2 → 3 → ( 2 → 3 → ( 2 → 3 → ( 2 → 3 → ( 2 → 3 → ( 2 → 3 → ( 2 → 3 → 12 ) ) ) ) ) ) = 2 ↑ ⋯ ⋯ ⋯ ⋯ ↑ ⏟ 3 2 ↑ ⋯ ⋯ ⋯ ↑ ⏟ 3 ⋮ ⏟ 2 ↑ ⋯ ⋯ ⋯ ↑ ⏟ 3 2 ↑ 12 3 } 7 layers = 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 12 3 3 3 3 3 3 3 = 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑↑↑↑↑↑↑↑↑↑↑↑ 3 3 3 3 3 3 3 = 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 → 3 → 12 3 3 3 3 3 3 = 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ hyper  ( 2 , 14 , 3 ) 3 3 3 3 3 3 {\displaystyle {\begin{aligned}F&:=F^{7}(12)=F\left(F\left(F\left(F\left(F\left(F\left(F(12)\right)\right)\right)\right)\right)\right)\\&=2\rightarrow 3\rightarrow \left(2\rightarrow 3\rightarrow \left(2\rightarrow 3\rightarrow \left(2\rightarrow 3\rightarrow \left(2\rightarrow 3\rightarrow \left(2\rightarrow 3\rightarrow \left(2\rightarrow 3\rightarrow 12\right)\right)\right)\right)\right)\right)\\&=\left.{\begin{matrix}2\underbrace {\uparrow \cdots \cdots \cdots \cdots \uparrow } 3\\2\underbrace {\uparrow \cdots \cdots \cdots \uparrow } 3\\\underbrace {\qquad \;\;\vdots \qquad \;\;} \\2\underbrace {\uparrow \cdots \cdots \cdots \uparrow } 3\\2\uparrow ^{12}3\end{matrix}}\right\}7{\text{ layers}}=2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{12}3}3}3}3}3}3}3=2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow 3}3}3}3}3}3}3\\&=2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\rightarrow 3\rightarrow 12}3}3}3}3}3}3\\&=2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{\operatorname {hyper} ({2,14,3})}3}3}3}3}3}3\end{aligned}}}
2 → 3 → ( 2 → 3 → ( 2 → 3 → ( 2 → 3 → ( 2 → 3 → ( 2 → 3 → ( 2 → 4 → 11 ) ) ) ) ) ) {\displaystyle 2\rightarrow 3\rightarrow \left(2\rightarrow 3\rightarrow \left(2\rightarrow 3\rightarrow \left(2\rightarrow 3\rightarrow \left(2\rightarrow 3\rightarrow \left(2\rightarrow 3\rightarrow \left(2\rightarrow 4\rightarrow 11\right)\right)\right)\right)\right)\right)}
これはどちらも 2 → 3 → 12 = 2 → 4 → 11 {\displaystyle 2\rightarrow 3\rightarrow 12=2\rightarrow 4\rightarrow 11} を根拠としている。
3 ↑ G 63 ( 4 ) − 1 3 ↑ G 63 ( 4 ) − 2 ⋯ 3 ↑ 5 3 ↑ 4 3 ↑ 3 3 ↑ 2 7625597484987 {\displaystyle 3\uparrow ^{G^{63}(4)-1}3\uparrow ^{G^{63}(4)-2}\cdots 3\uparrow ^{5}3\uparrow ^{4}3\uparrow ^{3}3\uparrow ^{2}7625597484987} ,
3 ↑ G 63 ( 4 ) − 1 3 ↑ G 63 ( 4 ) − 2 ⋯ 3 ↑ 6 3 ↑ 5 3 ↑ 4 3 ↑ 3 Cg  ( 3 ) {\displaystyle 3\uparrow ^{G^{63}(4)-1}3\uparrow ^{G^{63}(4)-2}\cdots 3\uparrow ^{6}3\uparrow ^{5}3\uparrow ^{4}3\uparrow ^{3}\operatorname {Cg} \left(3\right)} (CgはCG関数, Cg  ( n ) = n → n → n → ⋯ → n → n → n ⏟ 長さ n {\displaystyle \operatorname {Cg} (n)=\underbrace {n\rightarrow n\rightarrow n\rightarrow \dots \rightarrow n\rightarrow n\rightarrow n} _{{\text{長さ}}n}} )