Graham's number

Graham's number, named after Ronald Graham, is a large number that is an upper bound on the solution to a certain problem in Ramsey theory.

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Gardner described it in the "Mathematical Games" section of Scientific American in November 1977, writing that, "In an unpublished proof, Graham has recently established ... 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 himself. While Graham was trying to explain a result in Ramsey theory which he had derived with his collaborator B. L. 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 Ramsey-theory problem studied by Graham and Rothschild.[1]

Graham's number is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes' number and Moser's number. Indeed, like the last three of those numbers, the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies at least one Planck volume. Even power towers of the form $\scriptstyle a ^{ b ^{ c ^{ \cdot ^{ \cdot ^{ \cdot}}}}}$ are useless for this purpose, although it can be easily described by recursive formulas using Knuth's up-arrow notation or the equivalent, as was done by Graham. The last ten digits of Graham's number are ...2464195387.

Specific integers known to be far larger than Graham's number have since appeared in many serious mathematical proofs (e.g., in connection with Friedman's various finite forms of Kruskal's theorem).

Context

Example of a 2-colored 3-dimensional cube containing one single-coloured 4-vertex coplanar complete subgraph. The subgraph is shown below the cube. Note that this cube would contain no such subgraph if, for example, the bottom edge in the present subgraph were replaced by a blue edge – thus proving by counterexample that N* > 3.

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

Consider an n-dimensional hypercube, and connect each pair of geometric vertices, to obtain a complete graph on 2n graph 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 complete subgraph on 4 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 $\scriptstyle F^7(12) \;=\; F(F(F(F(F(F(F(12)))))))$, where $\scriptstyle F(n) \;=\; 2\uparrow^n 3$ in Knuth's up-arrow notation; the number is between 4 → 2 → 8 → 2 and 2 → 3 → 9 → 2 in Conway chained arrow notation.[2] This was reduced in 2013 via upper bounds on the Hales–Jewett number to $\scriptstyle N' \;=\; 2\;\uparrow\uparrow\;2\;\uparrow\uparrow\;2\;\uparrow\uparrow\;9$.[3] 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 N* are 13 ≤ N* ≤ N'.

Graham's number, G, is much larger than N: $\scriptstyle f^{64}(4)$, where $\scriptstyle f(n) \;=\; 3 \uparrow^n 3$. 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.[4]

Definition

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

$\left. \begin{matrix} G &=&3\underbrace{\uparrow \uparrow \cdots\cdots\cdots\cdots\cdots \uparrow}3 \\ & &3\underbrace{\uparrow \uparrow \cdots\cdots\cdots\cdots \uparrow}3 \\ & &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\ & &3\underbrace{\uparrow \uparrow \cdots\cdot\cdot \uparrow}3 \\ & &3\uparrow \uparrow \uparrow \uparrow3 \end{matrix} \right \} \text{64 layers}$

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,

$G = g_{64},\text{ where }g_1=3\uparrow\uparrow\uparrow\uparrow 3,\ g_n = 3\uparrow^{g_{n-1}}3,$

and where a superscript on an up-arrow indicates how many arrows are there. In other words, G is calculated in 64 steps: the first step is to calculate g1 with four up-arrows between 3s; the second step is to calculate g2 with g1 up-arrows between 3s; the third step is to calculate g3 with g2 up-arrows between 3s; and so on, until finally calculating G = g64 with g63 up-arrows between 3s.

Equivalently,

$G = f^{64}(4),\text{ where }f(n) = 3 \uparrow^n 3,$

and the superscript on f indicates an iteration of the function, e.g., f 4(n) = f(f(f(f(n)))). Expressed in terms of the family of hyperoperations $\scriptstyle \text{H}_0, \text{H}_1, \text{H}_2, \cdots$, the function f is the particular sequence $\scriptstyle f(n) \;=\; \text{H}_{n+2}(3,3)$, which is a version of the rapidly growing Ackermann function A(n,n). (In fact, $\scriptstyle f(n) \;>\; A(n,\, n)$ for all n.) The function f can also be expressed in Conway chained arrow notation as $\scriptstyle f(n) \;=\; 3 \rightarrow 3 \rightarrow n$, and this notation also provides the following bounds on G:

$3\rightarrow 3\rightarrow 64\rightarrow 2 < G < 3\rightarrow 3\rightarrow 65\rightarrow 2.\,$

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 (g1) of the rapidly growing 64-term sequence. First, in terms of tetration ($\scriptstyle \uparrow\uparrow$) alone:

$g_1 = 3 \uparrow \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow \uparrow (3 \uparrow \uparrow \uparrow 3) = 3 \uparrow\uparrow (3 \uparrow\uparrow (3 \uparrow\uparrow \ \dots \ (3 \uparrow\uparrow 3) \dots ))$

where the number of 3s in the expression on the right is

$3 \uparrow \uparrow \uparrow 3 \ = \ 3 \uparrow \uparrow (3 \uparrow \uparrow 3).$

Now each tetration ($\scriptstyle\uparrow\uparrow$) operation reduces to a "tower" of exponentiations ($\scriptstyle \uparrow$) according to the definition

$3 \uparrow\uparrow X \ = \ 3 \uparrow (3 \uparrow (3 \uparrow \dots (3 \uparrow 3) \dots )) \ = \ 3^{3^{\cdot^{\cdot^{\cdot^{3}}}}} \quad \text{where there are X 3s}.$

Thus,

$g_1 = 3 \uparrow\uparrow (3 \uparrow\uparrow (3 \uparrow\uparrow \ \dots \ (3 \uparrow\uparrow 3) \dots )) \quad \text{where the number of 3s is} \quad 3 \uparrow \uparrow (3 \uparrow \uparrow 3)$

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

$g_1 = \left. \begin{matrix}3^{3^{\cdot^{\cdot^{\cdot^{\cdot^{3}}}}}}\end{matrix} \right \} \left. \begin{matrix}3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}\end{matrix} \right \} \dots \left. \begin{matrix}3^{3^3}\end{matrix} \right \} 3 \quad \text{where the number of towers is} \quad \left. \begin{matrix}3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}\end{matrix} \right \} \left. \begin{matrix}3^{3^3}\end{matrix} \right \} 3$

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, g1 is computed by first calculating the number of towers, n = 3↑3↑3↑...↑3 (where the number of 3s is 3↑3↑3 = 7625597484987), and then computing the nth tower in the following sequence:

```      1st tower:  3
     
      2nd tower:  3↑3↑3 (number of 3s is 3) = 7625597484987
     
      3rd tower:  3↑3↑3↑3↑...↑3 (number of 3s is 7625597484987) = …
     
      ⋮
     
 g1 = nth tower:  3↑3↑3↑3↑3↑3↑3↑...↑3 (number of 3s is given by the n-1th 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 g1.

The magnitude of this first term, g1, 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 g1, is far greater than the number of Planck volumes (roughly 10185 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 = g64 is reached.

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 (10d 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 different possible values of 3↑3↑…3↑x when all but the rightmost d decimal digits are ignored
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 3$\scriptstyle\uparrow$3↑…3↑x mod 10d, 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[5] for computing these digits may be described as follows: let x = 3, then iterate, d times, the assignment x = 3x mod 10d. 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 10k)≡[GG](mod 10k).

k=t-1, where G(t):=3↑↑t.[6] It follows that, g63 ≪ k ≪ g64.

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
```

Notes

1. Lavrov, Mikhail; Lee, Mitchell; Mackey, John (2013). "Graham's Number is Less Than 2 ↑↑↑ 6" (PDF). arXiv:1304.6910. Retrieved 2013-04-29.
2. Martin Gardner (1977) "In which joining sets of points leads into diverse (and diverting) paths". Scientific American, November 1977

References

• Graham, R. L.; Rothschild, B. L. (1971). "Ramsey's Theorem for n-Parameter Sets". Transactions of the American Mathematical Society 159: 257–292. doi:10.2307/1996010. JSTOR 1996010.  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", Studies in Combinatorics, Rota, G.-G., ed., Mathematical Association of America, 17:80–99. On p. 90, in stating "the best available estimate" for the solution, the explicit formula for N is repeated from the 1971 paper.
• Gardner, Martin (November 1977). "Mathematical Games". Scientific American 237: 18–28. ; 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.
• Gardner, Martin (2001). The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. New York, NY: Norton. ISBN 0-393-02023-1.
• Exoo, Geoffrey (2003). "A Euclidean Ramsey Problem". Discrete and Computational Geometry 29 (2): 223–227. doi:10.1007/s00454-002-0780-5.  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:0811.1055v1 [math.CO].
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