Graham's number is a tremendously large finite number that is a proven upper bound to the solution of a certain problem in Ramsey theory. It is named after mathematician Ronald Graham who used the number as a simplified explanation of the upper bounds of the problem he was working on in conversations with popular science writer Martin Gardner. The number was published in the 1980 Guinness Book of World Records, which added to the popular interest in the number. Graham's number is much larger than any other number you can imagine. It is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume which equals to about . Even power towers of the form are insufficient for this purpose, although it can be described by recursive formulas using Knuth's up-arrow notation. Though too large to be computed in full, many of the last digits of Graham's number can be derived through simple algorithms.
Specific integers known to be far larger than Graham's number have since appeared in many serious mathematical proofs, for example, in connection with Harvey Friedman's various finite forms of Kruskal's theorem.
The last 400 digits are these:
Graham's number is connected to the following problem in Ramsey theory:
Connect each pair of geometric vertices of an -dimensional hypercube to obtain a complete graph on vertices. Color each of the edges of this graph either red or blue. What is the smallest value of for which every such coloring contains at least one single-colored complete subgraph on four coplanar vertices?
In 1971, Graham and Rothschild proved that this problem has a solution giving as a bound with being a large but explicitly defined number where in Knuth's up-arrow notation; the number is between 4 → 2 → 8 → 2 and 2 → 3 → 9 → 2 in Conway chained arrow notation. This was reduced in 2014 via upper bounds on the Hales-Jewett number to The lower bound of 6 was later improved to 11 by Geoff Exoo in 2003, and to 13 by Jerome Barkley in 2008. Thus, the best known bounds for are
Graham's number, is much larger than where This weaker upper bound for the problem, attributed to an unpublished work of Graham, was eventually published and named by Martin Gardner in Scientific American in November 1977. The number gained a degree of popular attention when Martin Gardner described it in the "Mathematical Games" section of Scientific American in November 1977, writing that Graham had recently established, in an unpublished proof, "a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof."
Knuth's up-arrow notation is a way of describing very large numbers. It's defined recursively, with the base case of repeated multiplication. So, for example, means . Therefore, the first level of up-arrow notation is just exponentiation, and we can write as
However, each additional up-arrow requires repeated application of the previous level of up-arrows. Thus, means or These numbers grow very, very quickly; is trillions of digits long.
Using Knuth's up-arrow notation, Graham's number is
where the number of arrows in each subsequent layer is specified by the value of the next layer below it; that is, where a superscript on an up-arrow indicates how many arrows there are. In other words, is calculated in 64 steps: the first step is to calculate with four up-arrows between 3s; the second step is to calculate with up-arrows between 3s; the third step is to calculate with up-arrows between 3s, and so on, until finally calculating G = with up-arrows between 3s.
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 of the rapidly growing 64-term sequence. First, in terms of tetration ( ↑↑ ) alone, where the number of 3s in the expression on the right is
Now each tetration ( ↑↑ ) operation reduces to a power tower ( ↑ ) according to the definition Thus, where the number of 3s is becomes, solely in terms of repeated "exponentiation towers," where the number of 3s in each tower, starting from the leftmost tower, is specified by the value of the next tower to the right. Anyway other specific integers known to be far larger than Graham's number have since appeared in many serious mathematical proofs, for example, in connection with Harvey Friedman's various finite forms of Kruskal's theorem.