# 葛立恆數

## 定義

${\displaystyle 3\rightarrow 3\rightarrow 64\rightarrow 2<}$葛立恆數${\displaystyle <3\rightarrow 3\rightarrow 65\rightarrow 2}$

${\displaystyle G=\underbrace {3\uparrow ^{3\uparrow ^{3\uparrow ^{\cdot ^{\cdot ^{\cdot ^{3\uparrow \uparrow \uparrow \uparrow 3}\cdot }\cdot }\cdot }3}3}3} _{64{\text{ layers}}}=\left.3\underbrace {\uparrow \uparrow \cdots \cdots \cdots \cdots \uparrow } _{\displaystyle 3\underbrace {\uparrow \uparrow \cdots \cdots \cdots \uparrow } _{\displaystyle \underbrace {\qquad \vdots \qquad } _{\displaystyle 3\underbrace {\uparrow \uparrow \cdots \uparrow } _{\displaystyle 3\uparrow \uparrow \uparrow \uparrow 3}3}}3}3\right\}64{\text{ layers}}}$

## 巨大的葛立恆數

${\displaystyle \underbrace {3[3[3[\cdots 3[3[3} _{64}[6]3+2]3+2]3\cdots ]3+2]3+2]3}$

${\displaystyle g_{1}=3[6]3=3[5](3[5]3)=\underbrace {3[4](3[4](3[4]\cdots (3[4](3[4]3} _{3[5]3}))\cdots ))}$

${\displaystyle =3[4]7625597484987=\underbrace {3[3]3[3]\cdots [3]3} _{7625597484987}=\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} _{7625597484987}}$迭代冪次
${\displaystyle g_{1}=3[6]3=3[5](3[5]3)=3[5]X=\underbrace {3[4]3[4]\cdots [4]3} _{X}=\left.\underbrace {3^{3^{.^{.^{.{3}}}}}} _{\underbrace {3^{3^{.^{.^{.{3}}}}}} _{\underbrace {\vdots } _{3}}}\right\}X}$

${\displaystyle f(n)=3[n+2]3}$

${\displaystyle g_{1}=f(4)=3[6]3=3[5]X=\underbrace {3[4]3[4]\cdots [4]3} _{X}=\left.\underbrace {3^{3^{.^{.^{.{3}}}}}} _{\underbrace {3^{3^{.^{.^{.{3}}}}}} _{\underbrace {\vdots } _{3}}}\right\}X}$

${\displaystyle g_{2}=f^{2}(4)=f(f(4))=3[g_{1}+2]3}$

${\displaystyle g_{3}=f(f^{2}(4))=3[g_{2}+2]3}$
${\displaystyle \vdots \vdots }$
${\displaystyle g_{63}=f(f^{62}(4))=3[g_{62}+2]3}$

${\displaystyle G=g_{64}=f^{64}(4)=\underbrace {f(f(\cdots f} _{64}(4)\cdots ))=3[g_{63}+2]3}$

${\displaystyle g_{n}=3[g_{n-1}+2]3}$

${\displaystyle f^{2}(n)=f(f(n)),\ f^{3}(n)=f(f(f(n))),}$ 以此類推。

## 葛立恆數最尾端的500位數字

...

02425 95069 50647 38395 65747 91365 19351 79833 45353 62521

43003 54012 60267 71622 67216 04198 10652 26316 93551 88780

38814 48314 06525 26168 78509 55526 46051 07117 20009 97092

91249 54437 88874 96062 88291 17250 63001 30362 29349 16080

25459 46149 45788 71427 83235 08292 42102 09182 58967 53560

43086 99380 16892 49889 26809 95101 69055 91995 11950 27887

17830 83701 83402 36474 54888 22221 61573 22801 01329 74509

27344 59450 43433 00901 09692 80253 52751 83328 98844 61508

94042 48265 01819 38515 62535 79639 96189 93967 90549 66380

03222 34872 39670 18485 18643 90591 04575 62726 24641 95387.

## 参考文献

1. ^ Graham's number records. Iteror.org. [2014-04-09]. （原始内容存档于2013-10-19）.
2. ^ Lavrov, Mikhail; Lee, Mitchell; Mackey, John. Improved upper and lower bounds on a geometric Ramsey problem. European Journal of Combinatorics. 2014, 42: 135–144. doi:10.1016/j.ejc.2014.06.003.
3. ^ Exoo, Geoffrey. A Euclidean Ramsey Problem. Discrete & Computational Geometry. 2003, 29 (2): 223–227. doi:10.1007/s00454-002-0780-5.Exoo將Graham與Rothschild提出的上界${\displaystyle N}$稱為「葛立恆數」，但這不是馬丁·加德納所說的「葛立恆數」${\displaystyle G}$
4. ^ Barkley, Jerome. Improved lower bound on an Euclidean Ramsey problem. 2008. [math.CO].
5. ^ 馬丁·加德納 (1977) "In which joining sets of points leads into diverse (and diverting) paths"页面存档备份，存于互联网档案馆）. Scientific American, November 1977