GRAHAM'S NUMBER

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'Graham's number', named after Ronald Graham, is often described as the largest number that has ever been seriously used in a mathematical proof. It is too large to express in scientific notation so it needs special notation ($G$) 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.

Contents

Graham's problem

Definition of Graham's number

Magnitude of Graham's number

See also

References

External links

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 $2^n$ 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 that lies 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 qualified 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 $G$ is defined as
$$
G = left . egin{matrix} 3 underbrace{ uparrow ldots uparrow } 3 \ underbrace{ dots } \ 3 uparrowuparrowuparrowuparrow 3 end{matrix}
ight } 64

Equivalently,
:$G\; =\; g\_\{64\}$ where $g\_1=3uparrowuparrowuparrowuparrow\; 3$, $g\_n\; =\; 3uparrow^\{g\_\{n-1\}\}3$
or
:$G\; =\; f^\{64\}(4)$ where $f(n)\; =\; ext\{hyper\}(3,n+2,3)$ and hyper() is the hyper operator.
Graham's number ''G'' itself cannot succinctly be expressed in Conway chained arrow notation, but $3$
ightarrow 3
ightarrow 64
ightarrow 2 < G < 3
ightarrow 3
ightarrow 65
ightarrow 2 , 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:
:$$
g_1
= 3 uparrow uparrow uparrow uparrow 3
= 3 uparrow uparrow uparrow (3 uparrow uparrow uparrow 3)
= 3 uparrow uparrow uparrow
left(
egin{matrix}
underbrace{3^{3^{cdot^{cdot^{cdot^{3}}}}}} & \
3^{3^3} & ext{threes}
end{matrix}

ight)
= left.
egin{matrix}
underbrace{3^{3^{cdot^{cdot^{cdot^{3}}}}}} & \
underbrace{3^{3^{cdot^{cdot^{cdot^{3}}}}}} & ext{threes} \
dots & dots \
underbrace{3^{3^{cdot^{cdot^{cdot^{3}}}}}} & ext{threes} \
3^{3^3} & ext{threes} \
end{matrix}

ight }
egin{matrix}
& \
underbrace{3^{3^{cdot^{cdot^{cdot^{3}}}}}} & mbox{ layers} \
3^{3^3} & ext{ threes}
end{matrix}

Note that the arrows are ; e.g., $3\; uparrow\; 3\; uparrow\; 3\; =\; 3\; uparrow\; (3\; uparrow\; 3)\; =\; 3\; uparrow\; 27\; =\; 7,625,597,484,987$.
This first term, ''g''₁, is already inconceivably 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

★ Hyper operator

★ Skewes' number

References

★ Penrose Tiles to Trapdoor Ciphers, , Martin, Gardner, , 1989, ISBN 0-88385-521-6

★ Ramsey's Theorem for n-Parameter Sets, Graham, R. L., , , Transactions of the American Mathematical Society, 1971

External links

★ "A Ramsey Problem on Hypercubes" by Geoff Exoo

★ Mathworld article on Graham's number

★ How to calculate Graham's number