How do I write Grahams number

Question

I found that Graham's number is :

So, can we say that it is equal to $3^x$ with $x$ is a power tower of 63 3's?

Answer 1

Knuth's up-arrow notation works as follows :

$a\uparrow b=a^b$

$a\uparrow \uparrow b=a\uparrow a\uparrow ...\uparrow a \uparrow a$ with $b$ $a's$

$a\uparrow \uparrow \uparrow b=a\uparrow \uparrow a \uparrow \uparrow ... \uparrow \uparrow a \uparrow \uparrow a$ with $b$ $a's$

and so on.

Note that the calculation is done from right , so $3\uparrow \uparrow 3=3\uparrow (3\uparrow 3)$, not $(3\uparrow 3)\uparrow 3$.

Now, Graham's number is defined as follows.

Notice the sequence

$G_0=4$ , $G_{n+1}=3\uparrow^{G(n)}3 $ for all $n\ge 0$

Then Graham's number is $G_{64}$.

Already $G_1=3\uparrow^4 3=3\uparrow \uparrow \uparrow\uparrow 3$ is so large that its magnitude cannot be comprehended. $G_2$ already has $G_1$ up-arrows, $G_3$ has $G_2$ up-arrows and so on.

Just to imagine how big $G_1$ already is : First imagine the number $$N:=3 \uparrow \uparrow 3 \uparrow \uparrow 3=3\uparrow \uparrow 3^{27}$$

This is a power tower of $3's$ with height $3^{27}$.

Now how to get $G_1$ :

Step $1$ : $3$

Step $2$ : $M_2 :$ a power tower with $3$ $3's$

Step $3$ : $M_3 :$ a power tower with $M_2$ $3's$

Step $4$ : $M_4 :$ a power tower with $M_3$ $3's$

and so on.

At step $N$, you reach $G_1$. It is already hard to describe how this number can be constructed. It is absolutely hopeless trying to comprend its size. Now you should get a feeling how insane big Graham's number is.