Graham’s Number

Graham’s number is a very, very big number that was discovered by a man called Ronald Graham. It is the answer to a problem in an area of mathematics called Ramsey theory, and is one of the biggest number ever used in a mathematics study.

Even if every number in the number was written in the tiniest writing possible, it would still be too big to fit in all the universe that scientists have seen so far – in other words: the universe is just too small a place to be able to write this number in.

Graham’s number is connected to the following problem in the branch of mathematics known as Ramsey theory: (Note that the symbol ^ is used to denote “to the power”)

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 either red or blue. What is the smallest value of n for which every such colouring contains at least one single-coloured 4-vertex planar complete subgraph?

Graham and Rothschild proved in 1971 that this problem has a solution, N*, and gave as a bounding estimate 6 ≤ N* ≤ N, with the upper bound N a particular, explicitly defined, very large number. In other words, the smallest possible value of N was thought to be 6 in 1971. However, this answer has been debunked. A possible solution points to the smallest value of N being at least 11, it may well be 12, but the answer lies between 11 and Graham’s Number (G).

To convey the difficulty of appreciating the enormous size of Graham’s number, it may be helpful to express—in terms of exponentiation alone—just the first term (g1) of the rapidly growing 64-term sequence.

(I) 3x3x3 is 3^3 is 27.

(II) 3^^3 is 7625597484987 you can think of this as 3 mutiplied by its self 3^3 times so 3x3x3x3x3….27 times.

(III.a) 3^^^3 is so huge, its digits would fill up the universe and beyond. it has 3638334640025 decimal digits – and this is only the start.

(III.b) 3^^^3 can also be represented as g1.

(IV) g2 is equal to 3^^^^^^^^^^^^^^^^^^^^^^…3 the number of arrows (^) in this number is g1 this means there are 3^^^3, arrows, or levels of exponents, in g2.

(V) g3 is equal to 3^^^^^^^^^^…3 the number of arrows is the value of g2 and so on.

(VI) g64 is equal to G, graham’s number.

Because of the Knuth up-arrow notation described here we know that the last ten numbers in Graham’s number are […] 2464195387. But the actual entire number (the remaining cyphers preceding these last ten numbers put together in one large number) is virtually infinitely longer.

The point of Graham’s number (G) is that it is the answer to the upper bound of N in this particular hypercube problem. Whatever the smallest value of N must be in this particular problem remains unclear. The answer probably lies between 11 and G. There is quite a margin of error to render faulty.