Graham's number

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Using Knuth’s arrow notation (1976), Graham’s number

G

, named after Ronald Graham, (as defined in Gardner’s Scientific American article) is

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

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,

$G=g_{64},{\text{ where }}g_{1}=3\uparrow \uparrow \uparrow \uparrow 3,\ g_{n}=3\uparrow ^{g_{n-1}}3,$

and where a superscript on an up-arrow indicates how many arrows there are.

Notice that

$3\,{\uparrow }^{4}\,3\,:=\,3\,{\uparrow \uparrow \uparrow \uparrow }\,3\,=\,3\,{\uparrow \uparrow \uparrow (3\,{\uparrow \uparrow \uparrow }\,}\,3)\,=\,3\,{\uparrow \uparrow (3\,{\uparrow \uparrow (3\,{\uparrow \uparrow \cdots (3\,{\uparrow \uparrow }}}}\,3)\cdots )),\,$

where the number of 3s in the expression on the RHS is 3 ↑↑ (3 ↑↑ 3), and 3 ↑↑ 3 = 7625597484987.

Sequences

A014222

a (0) = 0

; thereafter,

a (n + 1) = 3 a (n)

.

{0, 1, 3, 27, 7625597484987, ...}

References

• Gardner, Martin (November 1977). “Mathematical Games”. Scientific American 237: pp. 18–28. http://www.nature.com/scientificamerican/journal/v237/n5/pdf/scientificamerican1177-18.pdf. ; reprinted (revised) in Gardner (2001), cited below.
• Gardner, Martin (2001). The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. New York, NY: Norton. ISBN 0-393-02023-1.