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| {{About|the large number named after Ronald Graham|the investing term named after Benjamin Graham|Graham number}}
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| '''Graham's number''', named after [[Ronald Graham]], is a [[Large numbers|large number]] that is an upper bound on the solution to a problem in [[Ramsey theory]].
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| 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described it in the "Mathematical Games" section of ''[[Scientific American]]'' in November 1977, writing that, "In an unpublished proof, Graham has recently established ...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 C. Baez|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.<ref>{{cite web | url = https://plus.google.com/117663015413546257905/posts/KJTgfjkTZCQ | author = [[John C. Baez|John Baez]] | year = 2013 | title = A while back I told you about Graham's number...}}</ref>
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| 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 [[Numerical digit|digital representation]] of Graham's number, assuming that each digit occupies one [[Planck units#Derived Planck units|Planck volume]]. Even [[Tetration|power towers]] of the form <math>\scriptstyle a ^{ b ^{ c ^{ \cdot ^{ \cdot ^{ \cdot}}}}}</math> are beyond useless for this purpose, although it can be easily described by recursive formulas using [[Knuth's up-arrow notation]] or equivalent, as was done by Graham. The last ten digits of Graham's number are ...2464195387.
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| 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#Friedman's finite form|Kruskal's theorem]]).
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| ==Context==
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| [[Image:GrahamCube.svg|right|thumb|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.]]
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| Graham's number is connected to the following problem in [[Ramsey theory]]:
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| {{quote|<p>Connect each pair of [[vertex (geometry)|geometric vertices]] of an ''n''-dimensional [[hypercube]] to obtain a [[complete graph]] on 2<sup>''n''</sup> [[vertex (graph theory)|vertices]]. 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 four [[coplanar]] vertices?}}
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| 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 <math>\scriptstyle F^7(12) \;=\; F(F(F(F(F(F(F(12)))))))</math>, where <math>\scriptstyle F(n) \;=\; 2\uparrow^n 3</math> in [[Knuth's up-arrow notation]]; the number is between 4 → 2 → 8 → 2 and 2 → 3 → 9 → 2 in [[Conway chained arrow notation]].<ref>[http://iteror.org/big/Source/Graham-Gardner/GrahamsNumber.html]</ref> This was reduced in 2013 via upper bounds on the [[Hales–Jewett theorem|Hales–Jewett number]] to <math>\scriptstyle N' \;=\; 2\;\uparrow\uparrow\;2\;\uparrow\uparrow\;2\;\uparrow\uparrow\;9</math>.<ref>{{cite web|last1=Lavrov|first1=Mikhail|last2=Lee|first2=Mitchell|last3=Mackey|first3=John|title=Graham's Number is Less Than 2 ↑↑↑ 6|year=2013|url=http://arxiv.org/pdf/1304.6910v1.pdf|format=PDF|accessdate=2013-04-29|arxiv=1304.6910}}</ref> 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'''.
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| Graham's number, ''G'', is much larger than ''N'': <math>\scriptstyle f^{64}(4)</math>, where <math>\scriptstyle f(n) \;=\; 3 \uparrow^n 3</math>. 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.<ref>[[Martin Gardner]] (1977) [http://iteror.org/big/Source/Graham-Gardner/GrahamsNumber.html "In which joining sets of points leads into diverse (and diverting) paths"]. Scientific American, November 1977</ref>
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| ==Definition==
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| Using [[Knuth's up-arrow notation]], Graham's number ''G'' (as defined in Gardner's ''Scientific American'' article) is
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| :<math>
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| \left.
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| \begin{matrix}
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| G &=&3\underbrace{\uparrow \uparrow \cdots\cdots\cdots\cdots\cdots \uparrow}3 \\
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| & &3\underbrace{\uparrow \uparrow \cdots\cdots\cdots\cdots \uparrow}3 \\
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| & &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\
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| & &3\underbrace{\uparrow \uparrow \cdots\cdot\cdot \uparrow}3 \\
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| & &3\uparrow \uparrow \uparrow \uparrow3
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| \end{matrix}
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| \right \} \text{64 layers}
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| </math>
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| :
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| 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,
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| :<math>G = g_{64},\text{ where }g_1=3\uparrow\uparrow\uparrow\uparrow 3,\ g_n = 3\uparrow^{g_{n-1}}3,</math>
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| 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''<sub>1</sub> with four up-arrows between 3s; the second step is to calculate ''g''<sub>2</sub> with ''g''<sub>1</sub> up-arrows between 3s; the third step is to calculate ''g''<sub>3</sub> with ''g''<sub>2</sub> up-arrows between 3s; and so on, until finally calculating ''G'' = ''g''<sub>64</sub> with ''g''<sub>63</sub> up-arrows between 3s.
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| Equivalently,
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| :<math>G = f^{64}(4),\text{ where }f(n) = 3 \uparrow^n 3,</math>
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| and the superscript on ''f'' indicates an [[Iterated function|iteration of the function]], e.g., <big>''f''<sup> 4</sup>(''n'') = ''f''(''f''(''f''(''f''(''n''))))</big>. Expressed in terms of the family of [[hyperoperation]]s <math>\scriptstyle \text{H}_0, \text{H}_1, \text{H}_2, \cdots</math>, the function ''f'' is the particular sequence <math>\scriptstyle f(n) \;=\; \text{H}_{n+2}(3,3)</math>, which is a version of the rapidly growing [[Ackermann function]] ''A''(''n'',''n''). (In fact, <math>\scriptstyle f(n) \;>\; A(n,\, n)</math> for all ''n''.) The function ''f'' can also be expressed in [[Conway chained arrow notation#Graham's number|Conway chained arrow notation]] as <math>\scriptstyle f(n) \;=\; 3 \rightarrow 3 \rightarrow n</math>, and this notation also provides the following bounds on ''G'':
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| :<math> 3\rightarrow 3\rightarrow 64\rightarrow 2 < G < 3\rightarrow 3\rightarrow 65\rightarrow 2.\, </math>
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| ===Magnitude===
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| 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''<sub>1</sub>) of the rapidly growing 64-term sequence. First, in terms of [[tetration]] (<math>\scriptstyle \uparrow\uparrow</math>) alone:
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| :<math>
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| g_1
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| = 3 \uparrow \uparrow \uparrow \uparrow 3
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| = 3 \uparrow \uparrow \uparrow (3 \uparrow \uparrow \uparrow 3)
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| = 3 \uparrow\uparrow (3 \uparrow\uparrow (3 \uparrow\uparrow \ \dots \ (3 \uparrow\uparrow 3) \dots ))
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| </math>
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| where the number of 3s in the expression on the right is
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| :<math>3 \uparrow \uparrow \uparrow 3 \ = \ 3 \uparrow \uparrow (3 \uparrow \uparrow 3).</math>
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| Now each [[tetration]] (<math>\scriptstyle\uparrow\uparrow</math>) operation reduces to a "tower" of exponentiations (<math>\scriptstyle \uparrow</math>) according to the definition
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| :<math>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}.</math>
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| Thus,
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| :<math>
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| g_1
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| = 3 \uparrow\uparrow (3 \uparrow\uparrow (3 \uparrow\uparrow \ \dots \ (3 \uparrow\uparrow 3) \dots ))
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| \quad \text{where the number of 3s is}
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| \quad 3 \uparrow \uparrow (3 \uparrow \uparrow 3)
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| </math>
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| becomes, solely in terms of repeated "exponentiation towers",
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| :<math>
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| g_1 =
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| \left.
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| \begin{matrix}3^{3^{\cdot^{\cdot^{\cdot^{\cdot^{3}}}}}}\end{matrix}
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| \right \}
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| \left.
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| \begin{matrix}3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}\end{matrix}
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| \right \}
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| \dots
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| \left.
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| \begin{matrix}3^{3^3}\end{matrix}
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| \right \}
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| 3
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| \quad \text{where the number of towers is} \quad
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| \left.
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| \begin{matrix}3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}\end{matrix}
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| \right \}
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| \left.
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| \begin{matrix}3^{3^3}\end{matrix}
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| \right \}
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| 3
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| </math>
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| 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.
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| In other words, ''g''<sub>1</sub> 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 ''n''<sup>th</sup> tower in the following sequence:
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| 1st tower: <u>3</u>
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|
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| 2nd tower: 3↑3↑3 (number of 3s is <u>3</u>) = <u>7625597484987</u>
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|
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| 3rd tower: 3↑3↑3↑3↑...↑3 (number of 3s is <u>7625597484987</u>) = …
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|
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| ⋮
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|
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| ''g''<sub>1</sub> = ''n''<sup>th</sup> tower: 3↑3↑3↑3↑3↑3↑3↑...↑3 (number of 3s is given by the <u>''n-1''<sup>th</sup> tower</u>)
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| 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''<sub>1</sub>.
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| The magnitude of this first term, ''g''<sub>1</sub>, 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''<sub>1</sub>, is far greater than the number of [[Planck units|Planck volumes]] (roughly 10<sup>185</sup> 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''<sub>64</sub> is reached.
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| ==Rightmost decimal digits==
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| 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.
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| 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<sup>''d''</sup> 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:
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| {| class="wikitable" style="text-align:center" cellpadding="5"
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| |+ Number of different possible values of 3↑3↑…3↑''x'' when all but the rightmost ''d'' decimal digits are ignored <br>
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| ! width="150pt" | Number of digits (''d'') !! width="40pt" | 3↑''x'' !! width="60pt" | 3↑3↑''x'' !! width="80pt" | 3↑3↑3↑''x'' !! width="100pt" | 3↑3↑3↑3↑''x'' !! width="120pt" | 3↑3↑3↑3↑3↑''x''
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| |-
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| ! 1
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| | 4 <br> (1,3,9,'''7''') || 2 <br> (3,'''7''') || style="background:#05FFD3;" | 1 <br> ('''7''')|| style="background:#05FFD3;" | 1 <br> ('''7''') || style="background:#05FFD3;" | 1 <br> ('''7''')
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| |-
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| ! 2
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| | 20 <br> (01,03,…,'''87''',…,67) || 4 <br> (03,27,83,'''87''') || 2 <br> (27,'''87''') || style="background:#05FFD3;" | 1 <br> ('''87''') || style="background:#05FFD3;" | 1 <br> ('''87''')
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| |-
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| ! 3
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| | 100 <br> (001,003,…,'''387''',…,667) || 20 <br> (003,027,…'''387''',…,587) || 4 <br> (027,987,227,'''387''')|| 2 <br> (987,'''387''') || style="background:#05FFD3;" | 1 <br> ('''387''')
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| |}
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| 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<math>\scriptstyle\uparrow</math>3↑…3↑''x'' mod 10<sup>''d''</sup>, 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.
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| A simple algorithm<ref>[http://mathforum.org/library/drmath/view/51625.html The Math Forum @ Drexel ("Last Eight Digits of Z")]</ref> for computing these digits may be described as follows: let x = 3, then iterate, ''d'' times, the [[Assignment (computer science)|assignment]] ''x'' = 3<sup>''x''</sup> mod 10<sup>''d''</sup>. 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.)
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| Let ''k'' be the numerousness of these ''stable'' digits, which satisfy the congruence relation G(mod 10<sup>''k''</sup>)≡[G<sup>G</sup>](mod 10<sup>''k''</sup>).
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| ''k''=''t''-1, where G(''t''):=3↑↑''t''.<ref>[http://books.google.it/books?id=fdfjg7uVjJ0C&printsec=frontcover&hl=it#v=onepage&q&f=false Ripà, Marco (2011). ''La strana coda della serie n^n^…^n'', Trento, UNI Service. ISBN 978-88-6178-789-6]</ref> It follows that, {{nowrap|g<sub>63</sub> ≪ k ≪ g<sub>64</sub>}}.
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| The algorithm above produces the following 500 rightmost decimal digits of Graham's number (or of any tower of more than 500 3s):
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| …02425950695064738395657479136519351798334535362521
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| 43003540126026771622672160419810652263169355188780
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| 38814483140652526168785095552646051071172000997092
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| 91249544378887496062882911725063001303622934916080
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| 25459461494578871427832350829242102091825896753560
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| 43086993801689249889268099510169055919951195027887
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| 17830837018340236474548882222161573228010132974509
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| 27344594504343300901096928025352751833289884461508
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| 94042482650181938515625357963996189939679054966380
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| 03222348723967018485186439059104575627262464195387
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| ==See also==
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| * [[Ackermann function]]
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| * [[Kruskal's tree theorem]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{cite journal|author=Graham, R. L. |coauthors= Rothschild, B. L. |title=Ramsey's Theorem for n-Parameter Sets |journal= Transactions of the American Mathematical Society |volume=159 |pages=257–292 |year=1971 |doi=10.2307/1996010 |url=http://www.cs.umd.edu/~gasarch/vdw/Graham-Rothchild.pdf|jstor=1996010}} The explicit formula for ''N'' appears on p. 290. This is not the "Graham's number" ''G'' published by Martin Gardner.
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| * 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.
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| * {{cite journal|author=Gardner, Martin|title=Mathematical Games|journal= Scientific American|volume=237|pages=18–28|date=November 1977|url=http://www.nature.com/scientificamerican/journal/v237/n5/pdf/scientificamerican1177-18.pdf}}; reprinted (revised) in Gardner (2001), cited below.
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| * {{cite book|year=1989|title=Penrose Tiles to Trapdoor Ciphers|isbn=0-88385-521-6|first=Martin|last=Gardner |authorlink= Martin Gardner|publisher=Mathematical Association of America|location=Washington, D.C.}}
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| * {{cite book|year=2001|title=The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems|isbn=0-393-02023-1|first=Martin|last=Gardner |authorlink= Martin Gardner|publisher=Norton|location=New York, NY}}
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| * {{cite journal|author=Exoo, Geoffrey|title=A Euclidean Ramsey Problem|journal= Discrete and Computational Geometry|volume=29|pages=223–227|year=2003|doi=10.1007/s00454-002-0780-5|issue=2}} 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.
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| * {{cite arxiv|author=Barkley, Jerome|title=Improved lower bound on an Euclidean Ramsey problem|year=2008|eprint=0811.1055v1|class=math.CO }}
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| ==External links==
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| * "[http://isu.indstate.edu/ge/GEOMETRY/cubes.html A Ramsey Problem on Hypercubes]" by Geoff Exoo
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| * [http://mathworld.wolfram.com/GrahamsNumber.html Mathworld article on Graham's number]
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| * [http://www-users.cs.york.ac.uk/susan/cyc/g/graham.htm How to calculate Graham's number]
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| * [http://www.math.ucsd.edu/~fan/ron/papers/pre_cube.pdf Some Ramsey results for the n-cube] prepublication mentions Graham's number
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| * {{cite web|last=Padilla|first=Tony|title=Graham's Number|url=http://www.numberphile.com/videos/grahamsnumber.html|work=Numberphile|publisher=[[Brady Haran]]|coauthors=Parker, Matt}}
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| {{Large numbers}}
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| {{DEFAULTSORT:Graham's Number}}
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| [[Category:Ramsey theory]]
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| [[Category:Integers]]
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| [[Category:Large integers]]
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| [[Category:Large numbers]]
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| [[pl:Notacja strzałkowa#Liczba Grahama]]
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