グラハム数
2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 11 4 3 3 3 3 3 − 2 ( 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 11 4 3 3 3 3 3 − 3 ( 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 11 4 3 3 3 3 3 − 4 ( 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 11 4 3 3 3 3 3 − 5 ( ⋯ 2 ↑ 6 ( 2 ↑ 5 ( 2 ↑ 4 ( 2 ↑ 3 ( 2 ↑ 2 ( 4 ↑ 8 ) ) ) ) ) ⋯ ) ) ) ) {\displaystyle 2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{11}4}3}3}3}3}3-2}\left(2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{11}4}3}3}3}3}3-3}\left(2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{11}4}3}3}3}3}3-4}\left(2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{11}4}3}3}3}3}3-5}\left(\cdots 2\uparrow ^{6}\left(2\uparrow ^{5}\left(2\uparrow ^{4}\left(2\uparrow ^{3}\left(2\uparrow ^{2}\left(4\uparrow 8\right)\right)\right)\right)\right)\cdots \right)\right)\right)\right)}
2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 11 4 3 3 3 3 3 − 2 ( 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 11 4 3 3 3 3 3 − 3 ( 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 11 4 3 3 3 3 3 − 4 ( 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 2 ↑ 11 4 3 3 3 3 3 − 5 ( ⋯ 2 ↑ 6 ( 2 ↑ 5 ( 2 ↑ 4 ( 2 ↑ 3 ( 2 ↑ 2 ( ( 2 ↑ 2 ) ↑ ( 2 + ( 2 + 4 ) ) ) ) ) ) ) ⋯ ) ) ) ) {\displaystyle 2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{11}4}3}3}3}3}3-2}\left(2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{11}4}3}3}3}3}3-3}\left(2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{11}4}3}3}3}3}3-4}\left(2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{2\uparrow ^{11}4}3}3}3}3}3-5}\left(\cdots 2\uparrow ^{6}\left(2\uparrow ^{5}\left(2\uparrow ^{4}\left(2\uparrow ^{3}\left(2\uparrow ^{2}\left(\left(2\uparrow 2\right)\uparrow \left(2+\left(2+4\right)\right)\right)\right)\right)\right)\right)\cdots \right)\right)\right)\right)} ,
3 ↑ G 63 ( 4 ) − 1 3 ↑ G 63 ( 4 ) − 2 ⋯ 3 ↑ 5 3 ↑ 4 3 ↑ 3 3 ↑ 2 3 ↑ 1 { 3 × ( 3 + 6 ) } {\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}3\uparrow ^{1}\left\{3\times \left(3+6\right)\right\}} ,
3 ↑ G 63 ( 4 ) − 1 3 ↑ G 63 ( 4 ) − 2 ⋯ 3 ↑ 5 3 ↑ 4 3 ↑ 3 3 ↑ 2 3 3 ↑ 1 ( 3 + 6 ) {\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}3^{3}\uparrow ^{1}\left(3+6\right)} ,
3 ↑ G 63 ( 4 ) − 1 3 ↑ G 63 ( 4 ) − 2 ⋯ 3 ↑ 5 3 ↑ 4 3 ↑ 3 3 ↑ 2 27 ↑ 1 ( 3 + 6 ) {\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}27\uparrow ^{1}\left(3+6\right)} ,
3 ↑ G 63 ( 4 ) − 1 ( 3 ↑ G 63 ( 4 ) 2 ) , 3 ↑ G 63 ( 4 ) − 2 3 ↑ G 63 ( 4 ) − 1 ( 3 ↑ G 63 ( 4 ) 2 − 1 ) , 3 ↑ ( 3 ↑ 2 ( 3 ↑ 3 ( 3 ↑ 4 ( ⋯ 3 ↑ G 63 ( 4 ) − 2 ( 3 ↑ G 63 ( 4 ) − 1 ( 3 ↑ G 63 ( 4 ) − 1 3 − 1 ) − 1 ) ⋯ − 1 ) − 1 ) − 1 ) ) , 3 ↑ ( 3 ↑ 2 ( 3 ↑ 3 ( 3 ↑ 4 ( ⋯ 3 ↑ G 63 ( 4 ) − 2 ( 3 ↑ G 63 ( 4 ) − 1 ( G ( G 63 ( 4 ) − 1 ) − 1 ) − 1 ) ⋯ − 1 ) − 1 ) − 1 ) ) , 3 3 ↑ 2 ( 3 ↑ 3 ( 3 ↑ 4 ( ⋯ 3 ↑ G 63 ( 4 ) − 2 ( 3 ↑ G 63 ( 4 ) − 1 ( G ( G 63 ( 4 ) − 1 ) − 1 ) − 1 ) ⋯ − 1 ) − 1 ) − 1 ) ( ∵ G ( n ) = 3 ↑ n 3 = 3 ↑ n − 1 3 ↑ n − 1 3 = 3 ↑ n − 1 G ( n − 1 ) = 3 ↑ n − 2 3 ↑ n − 2 3 ↑ n − 2 3 ↑ n − 2 3 ↑ n − 2 3 ↑ n − 2 ⋯ ↑ n − 2 3 ↑ n − 2 3 ↑ n − 2 3 ⏟ G ( n − 1 ) 個の 3 = 3 ↑ n − 2 3 ↑ n − 2 3 ↑ n − 2 3 ↑ n − 2 3 ↑ n − 2 3 ↑ n − 2 ⋯ ↑ n − 2 3 ↑ n − 2 3 ↑ n − 2 3 ⏟ ( G ( n − 1 ) − 1 ) 個の 3 = 3 ↑ n − 2 ( 3 ↑ n − 1 ( G ( n − 1 ) − 1 ) ) ) {\displaystyle {\begin{aligned}&3\uparrow ^{G^{63}(4)-1}\left(3\uparrow ^{G^{63}(4)}2\right),\\&3\uparrow ^{G^{63}(4)-2}3\uparrow ^{G^{63}(4)-1}\left(3\uparrow ^{G^{63}(4)}2-1\right),\\&3\uparrow \left(3\uparrow ^{2}\left(3\uparrow ^{3}\left(3\uparrow ^{4}\left(\cdots 3\uparrow ^{G^{63}(4)-2}\left(3\uparrow ^{G^{63}(4)-1}\left(3\uparrow ^{G^{63}(4)-1}3-1\right)-1\right)\cdots -1\right)-1\right)-1\right)\right),\\&3\uparrow \left(3\uparrow ^{2}\left(3\uparrow ^{3}\left(3\uparrow ^{4}\left(\cdots 3\uparrow ^{G^{63}(4)-2}\left(3\uparrow ^{G^{63}(4)-1}\left(G\left(G^{63}(4)-1\right)-1\right)-1\right)\cdots -1\right)-1\right)-1\right)\right),\\&3^{3\uparrow ^{2}\left(3\uparrow ^{3}\left(3\uparrow ^{4}\left(\cdots 3\uparrow ^{G^{63}(4)-2}\left(3\uparrow ^{G^{63}(4)-1}\left(G\left(G^{63}(4)-1\right)-1\right)-1\right)\cdots -1\right)-1\right)-1\right)}\end{aligned}}\left({\begin{aligned}\because G(n)=3\uparrow ^{n}3=&3\uparrow ^{n-1}3\uparrow ^{n-1}3=3\uparrow ^{n-1}G(n-1)\\=&\underbrace {3\uparrow ^{n-2}3\uparrow ^{n-2}3\uparrow ^{n-2}3\uparrow ^{n-2}3\uparrow ^{n-2}3\uparrow ^{n-2}\cdots \uparrow ^{n-2}3\uparrow ^{n-2}3\uparrow ^{n-2}3} _{G(n-1){\text{個の}}3}\\=&3\uparrow ^{n-2}\underbrace {3\uparrow ^{n-2}3\uparrow ^{n-2}3\uparrow ^{n-2}3\uparrow ^{n-2}3\uparrow ^{n-2}\cdots \uparrow ^{n-2}3\uparrow ^{n-2}3\uparrow ^{n-2}3} _{\left(G(n-1)-1\right){\text{個の}}3}\\=&3\uparrow ^{n-2}\left(3\uparrow ^{n-1}\left(G(n-1)-1\right)\right)\end{aligned}}\right)}
F(x) を実際に計算してみると、
F(1) = 2↑3 = 2→3→1 = 2→3 = 2 = 8
F(2) = 2↑↑3 = 2→3→2 = 2↑(2↑2) = 2↑4 = 16
F(3) = 2↑↑↑3 = 2→3→3 = 2↑↑(2↑↑2) = 2↑↑4 = 2 = 2↑F(2) = 65536
F(4) = 2↑↑↑↑3 = 2→3→4 = 2↑↑↑(2↑↑↑2) = 2↑↑↑4 = 2↑↑2↑↑2↑↑2 = 2↑↑F(3) = 2↑↑65536
= 2 2 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 2 2 2 ⏟ 65536 {\displaystyle =\underbrace {2^{2^{2^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{2^{2^{2}}}}}}}}}}}}} _{65536}}
F(5) = 2↑↑↑↑↑3 = 2→3→5 = 2↑↑↑↑2↑↑↑↑2 = 2↑↑↑↑4 = 2↑↑↑2↑↑↑2↑↑↑2 = 2↑↑↑F(4)
= 2 ↑↑ 2 ↑↑ ⋯ ↑↑ 2 ↑↑ 2 ⏟ F ( 4 ) copies of 2 {\displaystyle =\underbrace {2\uparrow \uparrow 2\uparrow \uparrow \cdots \uparrow \uparrow 2\uparrow \uparrow 2} _{F(4){\text{ copies of }}2}}
:
:
F(12) = 2↑3 = 2→3→12 = 2↑2↑2 = 2↑4 = 2↑2↑2↑2 = 2↑F(11)
ハイパー演算表記