\begin{definition}[Definition:Graham's Number]
Let $3 \uparrow 3$ denote Knuth's notation for powers:
:$3 \uparrow 3 := 3^3$
Further, let:
:$3 \uparrow \uparrow 3 := 3 \uparrow \paren {3 \uparrow 3} = 3^{\paren {3^3} }$
and so define:
:$3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_n 3 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n - 1} \paren {3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n - 1} 3}$
Thus, for example:
:$3 \uparrow \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow \uparrow \paren {3 \uparrow \uparrow \uparrow 3}$
Let:
: $n_1 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{3 \uparrow \uparrow \uparrow \uparrow 3} 3$
That is, a total of $3 \uparrow \uparrow \uparrow \uparrow 3$ instances of $\uparrow$.
Similarly, let:
:$n_2 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_1} 3$
In general for $m \ge 2$:
:$n_m := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_{m - 1} } 3$
and so specifically:
:$n_{63} := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_{62} } 3$
It is this $n_{63}$ which is defined as '''Graham's number'''.
{{NamedforDef|Ronald Lewis Graham|cat = Graham}}
\end{definition}