167

\begin{eqnarray*} g_0 &=& 4 \\ g_1 &=& 3 \uparrow\uparrow\uparrow\uparrow 3 \\ g_2 &=& 3 \underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{g_1 \text{ 個箭號}} 3 \\ g_{k + 1} &=& 3 \underbrace{\uparrow\uparrow\uparrow\cdots\uparrow\uparrow\uparrow}_{g_k \text{ 個箭號}} 3 \ \ (\forall k \in \mathbb{N})\\ g_{64} &=& \text{葛立恆數} \end{eqnarray*}

## 近似

$$3 \rightarrow 3 \rightarrow 64 \rightarrow 2 < g_{64} < 3 \rightarrow 3 \rightarrow 65 \rightarrow 2$$ （鏈式箭號表示法）

$$\{3,65,1,2\} < g_{64} << \{3,66,1,2\}$$ （BEAF）

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