Graham's Number - Definition

Definition

Using Knuth's up-arrow notation, Graham's number G (as defined in Gardner's Scientific American article) is

\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\;\; \vdots \qquad\;\;} \\ & &3\underbrace{\uparrow \uparrow \cdots\cdot\cdot \uparrow}3 \\ & &3\uparrow \uparrow \uparrow \uparrow3 \end{matrix} \right \} \text{64 layers}

where the number of arrows in each layer, starting at the top layer, is specified by the value of the next layer below it; that is,

and where a superscript on an up-arrow indicates how many arrows are there. In other words, G is calculated in 64 steps: the first step is to calculate g1 with four up-arrows between 3s; the second step is to calculate g2 with g1 up-arrows between 3s; the third step is to calculate g3 with g2 up-arrows between 3s; and so on, until finally calculating G = g64 with g63 up-arrows between 3s.

Equivalently,

and the superscript on f indicates an iteration of the function, e.g., f 4(n) = f(f(f(f(n)))). Expressed in terms of the family of hyperoperations, the function f is the particular sequence, which is a version of the rapidly growing Ackermann function A(n,n). (In fact, for all n.) The function f can also be expressed in Conway chained arrow notation as, and this notation also provides the following bounds on G:

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