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Graham's number is one of the biggest numbers ever used in a constructive/practical way. It uses Knuth uparrow notation.

## Definition

Graham's number uses the G function, with ${\displaystyle G_{64}}$ being Graham's number.

${\displaystyle G_1}$ would be ${\displaystyle 3 \uparrow \uparrow \uparrow \uparrow 3}$. This number alone is ridiculously big, and is pretty much unimaginable. However, ${\displaystyle G_2}$ would be ${\displaystyle 3 \uparrow \cdots G_1 \cdots \uparrow 3}$. ${\displaystyle G_3}$ would be ${\displaystyle 3 \uparrow \cdots G_2 \cdots \uparrow 3}$, and so on until ${\displaystyle 3 \uparrow \cdots G_{63} \cdots \uparrow 3}$, which is ${\displaystyle G_{64}}$.

Graham's number can also be represented like this:

${\displaystyle \left. \begin{matrix} G &=&3\underbrace{\uparrow \uparrow \cdots \cdots \cdots \cdots \cdots \uparrow}3 \\ & &3\underbrace{\uparrow \uparrow \cdots \cdots \cdots \cdots \uparrow}3 \\ & & \underbrace{\qquad \quad \vdots \qquad \quad} \\ & &3\underbrace{\uparrow \uparrow \cdots \cdots \uparrow}3 \\ & &3\uparrow \uparrow \uparrow \uparrow3 \end{matrix} \right \} \text{64 layers} }$