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| [[Image:MarkoffNumberTree.png|thumb|450px|The first levels of the Markov number tree]]
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| A '''Markov number''' or '''Markoff number''' is a positive integer ''x'', ''y'' or ''z'' that is part of a solution to the Markov [[Diophantine equation]]
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| :<math>x^2 + y^2 + z^2 = 3xyz,\,</math>
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| studied by {{harvs|txt|authorlink=Andrey Markov|first=Andrey|last=Markoff|year1=1879|year2=1880}}.
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| The first few Markov numbers are | |
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| :[[1 (number)|1]], [[2 (number)|2]], [[5 (number)|5]], [[13 (number)|13]], [[29 (number)|29]], [[34 (number)|34]], [[89 (number)|89]], [[169 (number)|169]], [[194 (number)|194]], [[233 (number)|233]], 433, 610, 985, 1325, ... {{OEIS|id=A002559}}
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| appearing as coordinates of the Markov triples
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| :(1, 1, 1), (1, 1, 2), (1, 2, 5), (1, 5, 13), (2, 5, 29), (1, 13, 34), (1, 34, 89), (2, 29, 169), (5, 13, 194), (1, 89, 233), (5, 29, 433), (89, 233, 610), etc.
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| There are infinitely many Markov numbers and Markov triples.
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| == Markov tree ==
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| There are two simple ways to obtain a new Markov triple from an old one (''x'', ''y'', ''z''). First, one may permute the 3 numbers ''x'',''y'',''z'', so in particular one can normalize the triples so that ''x'' ≤ ''y'' ≤ ''z''. Second, if (''x'', ''y'', ''z'') is a Markov triple then by [[Vieta jumping]] so is (''x'', ''y'', 3''xy'' − ''z''). Applying this operation twice returns the same triple one started with. Joining each normalized Markov triple to the 1, 2, or 3 normalized triples one can obtain from this gives a graph starting from (1,1,1) as in the diagram. This graph is connected; in other words every Markov triple can be connected to (1,1,1) by a sequence of these operations.<ref>Cassels (1957) p.28</ref> If we start, as an example, with (1, 5, 13) we get its three neighbors (5, 13, 194), (1, 13, 34) and (1, 2, 5) in the Markov tree if ''x'' is set to 1, 5 and 13, respectively. For instance, starting with (1, 1, 2) and trading ''y'' and ''z'' before each iteration of the transform lists Markov triples with Fibonacci numbers. Starting with that same triplet and trading ''x'' and ''z'' before each iteration gives the triples with Pell numbers.
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| All the Markov numbers on the regions adjacent to 2's region are odd-indexed [[Pell number]]s (or numbers ''n'' such that 2''n''<sup>2</sup> − 1 is a square, {{OEIS2C|id=A001653}}), and all the Markov numbers on the regions adjacent to 1's region are odd-indexed [[Fibonacci number]]s ({{OEIS2C|id=A001519}}). Thus, there are infinitely many Markov triples of the form
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| :<math>(1, F_{2n - 1}, F_{2n + 1}),\,</math>
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| where ''F''<sub>''x''</sub> is the ''x''th Fibonacci number. Likewise, there are infinitely many Markov triples of the form
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| :<math>(2, P_{2n - 1}, P_{2n + 1}),\,</math>
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| where ''P''<sub>''x''</sub> is the ''x''th [[Pell number]].<ref>{{OEIS2C|id=A030452}} lists Markov numbers that appear in solutions where one of the other two terms is 5.</ref>
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| ==Other properties==
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| Aside from the two smallest ''singular'' triples (1,1,1) and (1,1,2), every Markov triple consists of three distinct integers.<ref>Cassels (1957) p.27</ref>
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| The ''unicity conjecture'' states that for a given Markov number ''c'', there is exactly one normalized solution having ''c'' as its largest element: proofs of this conjecture have been claimed but none seems to be correct.<ref>Guy (2004) p.263</ref>
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| Odd Markov numbers are 1 more than multiples of 4, while even Markov numbers are 2 more than multiples of 32.<ref>{{cite journal
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| | last = Zhang
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| | first = Ying
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| | title = Congruence and Uniqueness of Certain Markov Numbers
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| | journal = [[Acta Arithmetica]]
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| | volume = 128
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| | issue = 3
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| | year = 2007
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| | pages = 295–301
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| | url = http://journals.impan.gov.pl/aa/Inf/128-3-7.html
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| | mr = 2313995
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| | doi = 10.4064/aa128-3-7
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| | ref = Zhang2007}}</ref>
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| In his 1982 paper, [[Don Zagier]] conjectured that the ''n''th Markov number is asymptotically given by
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| :<math>m_n = \tfrac13 e^{C\sqrt{n}+o(1)} \quad\text{with } C = 2.3523418721 \ldots.</math>
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| Moreover he pointed out that <math>x^2 + y^2 + z^2 = 3xyz +4/9</math>, an extremely good approximation of the original Diophantine equation, is equivalent to <math>f(x)+f(y)=f(z)</math> with ''f''(''t'') = [[arcosh]](3''t''/2).<ref>{{cite journal
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| | last = Zagier
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| | first = Don B.
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| | title = On the Number of Markoff Numbers Below a Given Bound
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| | journal = [[Mathematics of Computation]]
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| | volume = 160
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| | year = 1982
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| | pages = 709–723
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| | doi = 10.2307/2007348
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| | mr = 0669663
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| | issue = 160
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| | jstor = 2007348
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| | ref = Zagier1982}}</ref> The conjecture was proved by [[Greg McShane]] and [[Igor Rivin]] in 1995 using techniques from hyperbolic geometry.<ref>{{cite journal
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| | author1 = Greg McShane
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| | author2 = Igor Rivin
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| | title = Simple curves on hyperbolic tori
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| | journal = C. R. Acad. Sci. Paris Sér. I. Math.
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| | volume = 320
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| | year = 1995
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| | number = 12
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| | ref = McShane1995}}</ref>
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| The ''n''th [[Lagrange number]] can be calculated from the ''n''th Markov number with the formula
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| :<math>L_n = \sqrt{9 - {4 \over {m_n}^2}}.\,</math>
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| ==Markov's theorem==
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| {{harvs|txt|last=Markoff|year1=1879|year2=1880}} showed that if
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| :<math>f(x,y) = ax^2+bxy+cy^2 \, </math>
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| is an indefinite binary quadratic form with real coefficients and [[discriminant]] <math>D = b^2-4ac</math>, then there are integers ''x'', ''y'' for which ''f'' takes a nonzero value of absolute value at most
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| :<math>\frac{\sqrt D}{3}</math>
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| unless ''f'' is a ''Markov form'':<ref>Cassels (1957) p.39</ref> a constant times a form
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| :<math>px^2+(3p-2a)xy+(b-3a)y^2 \, </math>
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| where (''p'', ''q'', ''r'') is a Markov triple and
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| :<math> 0<a<p/2,aq\equiv\pm r\bmod p, bp-a^2=1 \,</math>
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| ==Matrices==
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| If ''X'' and ''Y'' are in SL<sub>2</sub>('''C''') then
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| :<math> Tr(X)Tr(Y)Tr(XY) + Tr(XYX^{-1}Y^{-1})+2= Tr(X)^2+Tr(Y)^2+Tr(XY)^2 </math>
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| so that if Tr(''XYX''<sup>−1</sup>''Y''<sup>−1</sup>)=−2 then
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| :<math> Tr(X)Tr(Y)Tr(XY) = Tr(X)^2+Tr(Y)^2+Tr(XY)^2 </math>
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| In particular if ''X'' and ''Y'' also have integer entries then Tr(''X'')/3, Tr(''Y'')/3, and Tr(''XY'')/3 are a Markov triple. If ''XYZ'' = 1 then Tr(''XY'') = Tr(''Z''), so more symmetrically if ''X'', ''Y'', and ''Z'' are in SL<sub>2</sub>('''Z''') with ''XYZ'' = 1 and the commutator of two of them has trace −2, then their traces/3 are a Markov triple.
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| ==See also==
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| *[[Markov spectrum]]
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| == Notes ==
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| <references/>
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| ==References==
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| * {{cite book | first=J.W.S. | last=Cassels | authorlink=J. W. S. Cassels | title=An introduction to Diophantine approximation | series=Cambridge Tracts in Mathematics and Mathematical Physics | volume=45 | publisher=[[Cambridge University Press]] | year=1957 | zbl=0077.04801 }}
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| * {{cite book | first1=Thomas | last1=Cusick | first2=Mari | last2=Flahive | title=The Markoff and Lagrange spectra | series=Math. Surveys and Monographs | volume=30 | publisher=[[American Mathematical Society]] | location=Providence, RI | year=1989 | isbn=0-8218-1531-8 | zbl=0685.10023 }}
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| * {{cite book|first=Richard K. | last=Guy | authorlink=Richard K. Guy| title=[[Unsolved Problems in Number Theory]]| publisher=[[Springer-Verlag]]| year=2004|isbn=0-387-20860-7| zbl=1058.11001 | pages=263–265 }}
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| * {{eom|id=m/m062540|first=A.V.|last= Malyshev|title=Markov spectrum problem}}
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| * {{Cite journal | last1=Markoff | first1=A. | title=Sur les formes quadratiques binaires indéfinies | publisher=Springer Berlin / Heidelberg | journal=[[Mathematische Annalen]] | issn=0025-5831 }}
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| :: {{cite journal |title=First memory| journal=[[Mathematische Annalen]] | year=1879 | doi=10.1007/BF02086269 | volume=15 | pages=381–406 | issue=3–4 | ref=Markoff1879|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0015&DMDID=DMDLOG_0031&L=1 }}
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| :: {{cite journal |title=Second memory| journal=[[Mathematische Annalen]] | year=1880 | doi=10.1007/BF01446234 | volume=17 | pages=379–399 | issue=3 | ref=Markoff1880 |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0017&DMDID=DMDLOG_0045&L=1}}
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| [[Category:Diophantine equations]]
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| [[Category:Diophantine approximation]]
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| [[Category:Fibonacci numbers]]
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