グラハム数
グラハム数を表すにはいくつか等価な表現がある。
クヌースの矢印表記
3 ↑ G 63 ( 4 ) + 1 2 {\displaystyle 3\uparrow ^{G^{63}(4)+1}2} ,
3 ↑ G 63 ( 4 ) − 1 3 ↑ G 63 ( 4 ) − 1 3 {\displaystyle 3\uparrow ^{G^{63}(4)-1}3\uparrow ^{G^{63}(4)-1}3} ,
3 ↑ G 62 ( 3 ↑ 5 2 ) + 1 2 {\displaystyle 3\uparrow ^{G^{62}\left(3\uparrow ^{5}2\right)+1}2} ,
3 ↑ G 62 ( 3 ↑↑↑↑↑ 2 ) + 1 2 {\displaystyle 3\uparrow ^{G^{62}\left(3\uparrow \uparrow \uparrow \uparrow \uparrow 2\right)+1}2} ,
3 ↑ G 63 ( 4 ) − 1 3 ↑ G 63 ( 4 ) − 2 ⋯ 3 ↑ 5 3 ↑ 4 3 ↑ 3 3 ↑ 2 3 ↑ 1 [ 3 × { 3 + ( 3 + 3 ) } ] {\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+(3+3)\right\}\right]} ,
3 [ G 63 ( 3 [ 6 ] 3 ) + 2 ] 3 , 3 [ G 63 ( 3 [ 6 ] 3 ) + 3 ] 2 , 3 [ G 63 ( 3 [ 7 ] 2 ) + 2 ] 3 , 3 [ G 63 ( 3 [ 7 ] 2 ) + 3 ] 2 , H G 63 ( H 6 ( 3 , 3 ) ) + 2 ( 3 , 3 ) , H G 63 ( H 6 ( 3 , 3 ) ) + 3 ( 3 , 2 ) , H G 63 ( H 7 ( 3 , 2 ) ) + 2 ( 3 , 3 ) , H G 63 ( H 7 ( 3 , 2 ) ) + 3 ( 3 , 2 ) {\displaystyle 3\left[G^{63}(3[6]3)+2\right]3,~3\left[G^{63}(3[6]3)+3\right]2,~3\left[G^{63}(3[7]2)+2\right]3,~3\left[G^{63}(3[7]2)+3\right]2,~H_{G^{63}(H_{6}(3,3))+2}(3,3),~H_{G^{63}(H_{6}(3,3))+3}(3,2),~H_{G^{63}(H_{7}(3,2))+2}(3,3),~H_{G^{63}(H_{7}(3,2))+3}(3,2)} hyper  ( 3 , G 63 ( hyper  ( 3 , 6 , 3 ) ) + 2 , 3 ) , hyper  ( 3 , G 63 ( hyper  ( 3 , 6 , 3 ) ) + 3 , 2 ) , hyper  ( 3 , G 63 ( hyper  ( 3 , 7 , 2 ) ) + 2 , 3 ) , hyper  ( 3 , G 63 ( hyper  ( 3 , 7 , 2 ) ) + 3 , 2 ) {\displaystyle \operatorname {hyper} (3,G^{63}(\operatorname {hyper} (3,6,3))+2,3),~\operatorname {hyper} (3,G^{63}(\operatorname {hyper} (3,6,3))+3,2),~\operatorname {hyper} (3,G^{63}(\operatorname {hyper} (3,7,2))+2,3),~\operatorname {hyper} (3,G^{63}(\operatorname {hyper} (3,7,2))+3,2)}