# Graham's number

**Graham's number** is large number that is an upper bound on the solution to 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. Gardner later described the number in *Scientific American* in 1977, introducing it to the general public. 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 many other large numbers such as a googol, Skewes' number and Moser's number. As with all these, 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, the smallest possible volume known to modern physics. 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 or equivalent, as was done by Graham. Though too large to be computed in full, many of the last digits of Graham's number can be derived through simple algorithms. The last 12 digits are ...262464195387.

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.

## Contents

## Context

Graham's number is connected to the following problem in Ramsey theory:

Connect each pair of geometric vertices of an

n-dimensional hypercube to obtain a complete graph on 2^{n}vertices. Colour each of the edges of this graph either red or blue. What is the smallest value ofnfor whicheverysuch colouring contains at least one single-coloured complete subgraph on four coplanar vertices?

In 1971, Graham and Rothschild proved that this problem has a solution *N**, giving as a bound 6 ≤ *N** ≤ *N*, with *N* 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.^{[1]} This was reduced in 2014 via upper bounds on the Hales–Jewett number to .^{[2]} The lower bound of 6 was later improved to 11 by Geoff Exoo in 2003,^{[3]} and to 13 by Jerome Barkley in 2008.^{[4]} Thus, the best known bounds for *N** are 13 ≤ *N** ≤ *N'*.

Graham's number, *G*, is much larger than *N*: , 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.^{[5]}

## Publication

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." The 1980 *Guinness Book of World Records* repeated Gardner's claim, adding to the popular interest in this number. According to physicist John Baez, Graham invented the quantity now known as Graham's number in conversation with Gardner. While Graham was trying to explain a result in Ramsey theory which he had derived with his collaborator Bruce Lee Rothschild, Graham found that the quantity now known as Graham's number was easier to explain than the actual number appearing in the proof. Because the number which Graham described to Gardner is larger than the number in the paper itself, both are valid upper bounds for the solution to the problem studied by Graham and Rothschild.^{[6]}

## Definition

Using Knuth's up-arrow notation, Graham's number *G* (as defined in Gardner's *Scientific American* article) is

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,

and where a superscript on an up-arrow indicates how many arrows there are. In other words, *G* is calculated in 64 steps: the first step is to calculate *g*_{1} with four up-arrows between 3s; the second step is to calculate *g*_{2} with *g*_{1} up-arrows between 3s; the third step is to calculate *g*_{3} with *g*_{2} up-arrows between 3s; and so on, until finally calculating *G* = *g*_{64} with *g*_{63} up-arrows between 3s.

Equivalently,

and the superscript on *f* indicates an iteration of the function, e.g., . Expressed in terms of the family of hyperoperations , the function *f* is the particular sequence , which is a version of the rapidly growing Ackermann function *A*(*n*,*n*). (In fact, for all *n*.) The function *f* can also be expressed in Conway chained arrow notation as , and this notation also provides the following bounds on *G*:

### Magnitude

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 (*g*_{1}) 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,

becomes, solely in terms of repeated "exponentiation towers",

and 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.

In other words, *g*_{1} is computed by first calculating the number of towers, (where the number of 3s is ), and then computing the *n*^{th} tower in the following sequence:

1st tower:32nd tower: 3↑3↑3 (number of 3s is3) =76255974849873rd tower: 3↑3↑3↑3↑...↑3 (number of 3s is7625597484987) = … ⋮g_{1}=n^{th}tower: 3↑3↑3↑3↑3↑3↑3↑...↑3 (number of 3s is given by the)n-1^{th}tower

where the number of 3s in each successive tower is given by the tower just before it. Note that the result of calculating the third tower is the value of *n*, the number of towers for *g*_{1}.

The magnitude of this first term, *g*_{1}, is so large that it is practically incomprehensible, even though the above display is relatively easy to comprehend. Even *n*, the mere *number of towers* in this formula for *g*_{1}, is far greater than the number of Planck volumes (roughly 10^{185} of them) into which one can imagine subdividing the observable universe. And after this first term, still another 63 terms remain in the rapidly growing *g* sequence before Graham's number *G* = *g*_{64} is reached. To illustrate just how fast this sequence grows, while *g*_{1} is equal to s with only four up arrows, the number of up arrows in *g*_{2} is this incomprehensibly large number *g*_{1}.

## Rightmost decimal digits

Graham's number is a "power tower" of the form 3↑↑*n* (with a very large value of *n*), so its rightmost decimal digits must satisfy certain properties common to all such towers. One of these properties is that *all such towers of height greater than d (say), have the same sequence of d rightmost decimal digits*. This is a special case of a more general property: The *d* rightmost decimal digits of all such towers of height greater than *d*+2, are *independent* of the topmost "3" in the tower; i.e., the topmost "3" can be changed to any other nonnegative integer without affecting the *d* rightmost digits.

The following table illustrates, for a few values of *d*, how this happens. For a given height of tower and number of digits *d*, the full range of *d*-digit numbers (10^{d} of them) does *not* occur; instead, a certain smaller subset of values repeats itself in a cycle. The length of the cycle and some of the values (in parentheses) are shown in each cell of this table:

Number of digits (d) |
3↑x |
3↑3↑x |
3↑3↑3↑x |
3↑3↑3↑3↑x |
3↑3↑3↑3↑3↑x |
---|---|---|---|---|---|

1 | 4 (1,3,9, 7) |
2 (3, 7) |
1 ( 7) |
1 ( 7) |
1 ( 7) |

2 | 20 (01,03,…, 87,…,67) |
4 (03,27,83, 87) |
2 (27, 87) |
1 ( 87) |
1 ( 87) |

3 | 100 (001,003,…, 387,…,667) |
20 (003,027,… 387,…,587) |
4 (027,987,227, 387) |
2 (987, 387) |
1 ( 387) |

The particular rightmost *d* digits that are ultimately shared by all sufficiently tall towers of 3s are in bold text, and can be seen developing as the tower height increases. For any fixed number of digits *d* (row in the table), the number of values possible for 33↑…3↑*x* mod 10^{d}, as *x* ranges over all nonnegative integers, is seen to decrease steadily as the height increases, until eventually reducing the "possibility set" to a single number (colored cells) when the height exceeds *d*+2.

A simple algorithm^{[7]} for computing these digits may be described as follows: let x = 3, then iterate, *d* times, the assignment *x* = 3^{x} mod 10^{d}. Except for omitting any leading 0s, the final value assigned to *x* (as a base-ten numeral) is then composed of the *d* rightmost decimal digits of 3↑↑*n*, for all *n* > *d*. (If the final value of *x* has fewer than *d* digits, then the required number of leading 0s must be added.)

Let *k* be the numerousness of these *stable* digits, which satisfy the congruence relation G(mod 10^{k})≡[G^{G}](mod 10^{k}).

*k*=*t*-1, where G(*t*):=3↑↑*t*.^{[8]} It follows that, g_{63} ≪ k ≪ g_{64}.

The algorithm above produces the following 500 rightmost decimal digits of Graham's number (or of any tower of more than 500 3s):

…02425950695064738395657479136519351798334535362521 43003540126026771622672160419810652263169355188780 38814483140652526168785095552646051071172000997092 91249544378887496062882911725063001303622934916080 25459461494578871427832350829242102091825896753560 43086993801689249889268099510169055919951195027887 17830837018340236474548882222161573228010132974509 27344594504343300901096928025352751833289884461508 94042482650181938515625357963996189939679054966380 03222348723967018485186439059104575627262464195387

## References

### Notes

- ↑ "Graham's number records". Iteror.org. Retrieved 2014-04-09.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Lavrov, Mikhail; Lee, Mitchell; Mackey, John (2014). "Improved upper and lower bounds on a geometric Ramsey problem".
*European Journal of Combinatorics*.**42**: 135–144. doi:10.1016/j.ejc.2014.06.003.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Exoo, Geoffrey (2003). "A Euclidean Ramsey Problem".
*Discrete & Computational Geometry*.**29**(2): 223–227. doi:10.1007/s00454-002-0780-5.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> Exoo refers to the Graham & Rothschild upper bound*N*by the term "Graham's number". This is not the "Graham's number"*G*published by Martin Gardner. - ↑ Barkley, Jerome (2008). "Improved lower bound on an Euclidean Ramsey problem". arXiv:0811.1055 [math.CO].<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Martin Gardner (1977) "In which joining sets of points leads into diverse (and diverting) paths". Scientific American, November 1977
- ↑ John Baez (2013). "A while back I told you about Graham's number..."<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "The Math Forum @ Drexel ("Last Eight Digits of Z")". Mathforum.org. Retrieved 2014-04-09.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑
*Ripà, Marco (2011). ''La strana coda della serie n^n^…^n'', Trento, UNI Service. ISBN 978-88-6178-789-6*. Books.google.it. Retrieved 2014-04-09.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>

### Bibliography

- Gardner, Martin (November 1977). "Mathematical Games" (PDF).
*Scientific American*.**237**: 18–28. doi:10.1038/scientificamerican1177-18.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>; reprinted (revised) in Gardner (2001), cited below. - Gardner, Martin (1989).
*Penrose Tiles to Trapdoor Ciphers*. Washington, D.C.: Mathematical Association of America. ISBN 0-88385-521-6.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - Gardner, Martin (2001).
*The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems*. New York, NY: Norton. ISBN 0-393-02023-1.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - Graham, R. L.; Rothschild, B. L. (1971). "Ramsey's Theorem for n-Parameter Sets" (PDF).
*Transactions of the American Mathematical Society*.**159**: 257–292. doi:10.2307/1996010. JSTOR 1996010.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> The explicit formula for*N*appears on p. 290. This is not the "Graham's number"*G*published by Martin Gardner. - Graham, R. L.; Rothschild, B. L. (1978). "Ramsey Theory". In Rota, G-C (ed.).
*Studies in Combinatorics (MAA Studies in Mathematics)*.**17**. Mathematical Association of America. pp. 80–99. ISBN 0-88385-117-2.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> On p. 90, in stating "the best available estimate" for the solution, the explicit formula for*N*is repeated from the 1971 paper.

## External links

- Sbiis Saibian's article on Graham's number
- "A Ramsey Problem on Hypercubes" by Geoff Exoo
- Mathworld article on Graham's number
- How to calculate Graham's number
- Some Ramsey results for the n-cube prepublication mentions Graham's number
- Padilla, Tony; Parker, Matt. "Graham's Number".
*Numberphile*. Brady Haran.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - Ron Graham. "What is Graham's Number? (feat Ron Graham)" (video).
*Numberphile*. Brady Haran.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - Ron Graham. "How Big is Graham's Number? (feat Ron Graham)" (video).
*Numberphile*. Brady Haran.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>