Graham's number, named after Ronald Graham, is a large number often described as the largest finite number that has ever been seriously used in a mathematical proof. Guinness World Records even listed Graham's number as the World Champion largest number. It is too large to be written in scientific notation because even the digits in the exponent would exceed the number of atoms in the observable universe so it needs its own special notation (<math>G</math>) to write down. Graham's number is much larger than other well known large numbers such as a googol and a googolplex, and even larger than Moser's number, another well-known large number.
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Graham's problem
Graham's number is connected to the following problem in the branch of mathematics known as Ramsey theory:
- Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on <math>2^n</math> vertices. Then colour each of the edges of this graph using only the colours red and black. What is the smallest value of n for which every possible such colouring must necessarily contain a single-coloured complete sub-graph with 4 vertices which lie in a plane?
Although the solution to this problem is not yet known, Graham's number is the smallest known upper bound for it. This bound was found by Graham and B. L. Rothschild (see (GR), corollary 12). They also provided the lower bound 6, adding the qualifying understatement: "Clearly, there is some room for improvement here." In Penrose Tiles to Trapdoor Ciphers, Martin Gardner wrote, "Ramsey-theory experts believe the actual Ramsey number for this problem is probably 6, making Graham's number perhaps the worst smallest-upper-bound ever discovered." More recently Geoff Exoo of Indiana State University has shown (in 2003) that it must be at least 11 and provided evidence that it is larger.
Definition of Graham's number
Using Knuth's up-arrow notation, Graham's number <math>G</math> is defined as
- <math>
G = \left . \begin{matrix} 3 \underbrace{ \uparrow \ldots \uparrow } 3 \\ \underbrace{ \vdots } \\ 3 \uparrow\uparrow\uparrow\uparrow 3 \end{matrix} \right \} \text{64 layers} </math> Equivalently,
- <math>G = g_{64}</math> where <math>g_1=3\uparrow\uparrow\uparrow\uparrow 3</math>, <math>g_n = 3\uparrow^{g_{n-1}}3</math>
or
- <math>G = f^{64}(4)</math> where <math>f(n) = \text{hyper}(3,n+2,3)</math> and hyper() is the hyper operator.
Graham's number G itself cannot succinctly be expressed in Conway chained arrow notation, but <math> 3\rightarrow 3\rightarrow 64\rightarrow 2 < G < 3\rightarrow 3\rightarrow 65\rightarrow 2 </math>, see bounds on Graham's number in terms of Conway chained arrow notation.
Magnitude of Graham's number
Since appreciation of the true size of Graham's number can be difficult, it can be helpful to express the first term of the sequence in terms of exponentiation:
- <math>
g_1 = 3 \uparrow \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow \uparrow (3 \uparrow \uparrow \uparrow 3) = 3 \uparrow \uparrow \uparrow
\left(
\underbrace{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}}_{3^{3^3}\text{copies of 3}}
\right)
= \left.
\begin{matrix}
\underbrace{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}}_{\underbrace{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}}_{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}\text{copies of 3}}\text{copies of 3}} \\
\vdots \\
\underbrace{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}}_{\underbrace{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}}_{3^{3^3}\text{copies of 3}}\text{copies of 3}}\text{copies of 3}
\end{matrix}
\right \}
\left. \underbrace{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}}_{3^{3^3}\text{copies of 3}}
\right. \text{layers}
</math> Note that the arrows are right-associative; e.g., <math>3 \uparrow 3 \uparrow 3 = 3 \uparrow (3 \uparrow 3) = 3 \uparrow 27 = 7,625,597,484,987</math>. This first term, g1, is already much greater than the number of atoms in the observable universe, and grows at an enormous rate as it is iterated through the sequence g.
See also
References
- Gardner, Martin (1989). Penrose Tiles to Trapdoor Ciphers. ISBN 0-88385-521-6.
- Graham, R. L.; Rothschild, B. L. (1971). "Ramsey's Theorem for n-Parameter Sets". Transactions of the American Mathematical Society 159: 257-292.
