# Markov number

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{{Short description|1=Solution to x*x + y*y + z*z = 3xyz}}
{{distinguish|Markov constant|Markov theorem}}
A '''Markov number''' or '''Markoff number''' is a positive [integer](/source/integer) ''x'', ''y'' or ''z'' that is part of a solution to the '''Markov [Diophantine equation](/source/Diophantine_equation)'''
:<math>x^2 + y^2 + z^2 = 3xyz,\,</math>
studied by {{harvs|txt|authorlink=Andrey Markov|first=Andrey|last=Markoff|year1=1879|year2=1880}}.

The first few Markov numbers are
:[1](/source/1_(number)), [2](/source/2_(number)), [5](/source/5_(number)), [13](/source/13_(number)), [29](/source/29_(number)), [34](/source/34_(number)), [89](/source/89_(number)), [169](/source/169_(number)), [194](/source/194_(number)), [233](/source/233_(number)), 433, 610, 985, 1325, ... {{OEIS|id=A002559}}
appearing as coordinates of the Markov triples
:(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), (1, 233, 610), (2, 169, 985), (13, 34, 1325), ...
There are infinitely many Markov numbers and Markov triples.

== Markov tree ==
thumb|450px|The first levels of the Markov number tree
There are two simple ways to obtain a new Markov triple from an old one (''x'',&nbsp;''y'',&nbsp;''z''). First, one may [permute](/source/permutation) the 3 numbers ''x'',''y'',''z'', so in particular one can normalize the triples so that ''x''&nbsp;≤&nbsp;''y''&nbsp;≤&nbsp;''z''.  Second, if (''x'',&nbsp;''y'',&nbsp;''z'') is a Markov triple then so is (''x'',&nbsp;''y'',&nbsp;3''xy''&nbsp;−&nbsp;''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](/source/connectivity_(graph_theory)); in other words every Markov triple can be connected to {{nowrap|(1,1,1)}} by a sequence of these operations.<ref>Cassels (1957) p.28</ref>  If one starts, as an example, with {{nowrap|(1, 5, 13)}} we get its three [neighbors](/source/neighbourhood_(graph_theory)) {{nowrap|(5, 13, 194)}}, {{nowrap|(1, 13, 34)}} and {{nowrap|(1, 2, 5)}} in the Markov tree if ''z'' is set to 1, 5 and 13, respectively.  For instance, starting with {{nowrap|(1, 1, 2)}} and trading ''y'' and ''z'' before each iteration of the transform lists Markov triples with [Fibonacci number](/source/Fibonacci_number)s. Starting with that same triplet and trading ''x'' and ''z'' before each iteration gives the triples with [Pell number](/source/Pell_number)s.

All the Markov numbers on the regions adjacent to 2's region are [odd](/source/parity_(mathematics))-indexed Pell numbers (or numbers ''n'' such that 2''n''<sup>2</sup>&nbsp;−&nbsp;1 is a [square](/source/square_number), {{OEIS2C|id=A001653}}), and all the Markov numbers on the regions adjacent to 1's region are odd-indexed Fibonacci numbers ({{OEIS2C|id=A001519}}). Thus, there are infinitely many Markov triples of the form

:<math>(1, F_{2n-1}, F_{2n+1}),\,</math>

where ''F''<sub>''k''</sub> is the ''k''th [Fibonacci number](/source/Fibonacci_number). Likewise, there are infinitely many Markov triples of the form

:<math>(2, P_{2n-1}, P_{2n+1}),\,</math>

where ''P''<sub>''k''</sub> is the ''k''th [Pell number](/source/Pell_number).<ref>{{OEIS2C|id=A030452}} lists Markov numbers that appear in solutions where one of the other two terms is&nbsp;5.</ref>

==Other properties==
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>

The ''unicity conjecture'', as remarked by [Frobenius](/source/Ferdinand_Georg_Frobenius) in 1913,<ref name="Frobenius 1913 ">{{cite journal | last=Frobenius | first=G. | title=Über die Markoffschen Zahlen | journal=S. B. Preuss Akad. Wiss. | date=1913 | pages=458–487}}</ref> states that for a given Markov number ''c'', there is exactly one normalized solution having ''c'' as its largest element: [proofs](/source/mathematical_proof) of this [conjecture](/source/conjecture) have been claimed but none seems to be correct.<ref>Guy (2004) p.263</ref> [Martin Aigner](/source/Martin_Aigner)<ref>Aigner (2013)</ref> examines several weaker variants of the unicity conjecture. His fixed numerator conjecture was proved by Rabideau and Schiffler in 2020,<ref name="Rabideau Schiffler 2020 ">{{cite journal | last=Rabideau | first=Michelle | last2=Schiffler | first2=Ralf | title=Continued fractions and orderings on the Markov numbers | journal=[Advances in Mathematics](/source/Advances_in_Mathematics) | volume=370 | date=2020 | doi=10.1016/j.aim.2020.107231 | article-number=107231| arxiv=1801.07155 }}</ref> while the fixed denominator conjecture and fixed sum conjecture were proved by Lee, Li, Rabideau and Schiffler in 2023.<ref name="Lee Li Rabideau Schiffler 2023 ">{{cite journal | last=Lee | first=Kyungyong | last2=Li | first2=Li | last3=Rabideau | first3=Michelle | last4=Schiffler | first4=Ralf | title=On the ordering of the Markov numbers | journal=Advances in Applied Mathematics | volume=143 | date=2023 | doi=10.1016/j.aam.2022.102453 | article-number=102453| doi-access=free }}</ref>

None of the prime divisors of a Markov number is congruent to 3 modulo 4, which implies that an odd Markov number is 1 more than a multiple of 4.<ref>Aigner (2013) p. 55</ref> Furthermore, if <math>m</math> is a Markov number then none of the prime divisors of <math>9m^2-4</math> is congruent to 3 modulo 4. An [even](/source/parity_(mathematics)) Markov number is 2 more than a multiple of 32.<ref>{{cite journal
 | last = Zhang
 | first = Ying
 | title = Congruence and Uniqueness of Certain Markov Numbers
 | journal = [Acta Arithmetica](/source/Acta_Arithmetica)
 | volume = 128
 | issue = 3
 | year = 2007
 | pages = 295–301
 | url = http://journals.impan.gov.pl/aa/Inf/128-3-7.html
 | mr = 2313995
 | doi = 10.4064/aa128-3-7
 | ref = Zhang2007| arxiv = math/0612620
 | bibcode = 2007AcAri.128..295Z
 | s2cid = 9615526
 }}</ref>

In his 1982 paper, [Don Zagier](/source/Don_Zagier) conjectured that the ''n''th Markov number is asymptotically given by
:<math>m_n = \tfrac13 e^{C\sqrt{n+o(1)}} \quad\text{with } C = 2.3523414972 \ldots\,.</math>
The error <math>o(1) = (\log(3m_n)/C)^2 - n</math> is plotted below.

thumb|300px|Error in the approximation of large Markov numbers

Moreover, he pointed out that <math>x^2 + y^2 + z^2 = 3xyz + 4/9</math>, an approximation of the original Diophantine equation, is equivalent to <math>f(x)+f(y)=f(z)</math> with <math>f(t) =</math> arcosh<math>(3t/2).</math><ref>{{cite journal
 | last = Zagier
 | first = Don B.
 | title = On the Number of Markoff Numbers Below a Given Bound
 | journal = [Mathematics of Computation](/source/Mathematics_of_Computation)
 | volume = 160
 | year = 1982
 | pages = 709–723
 | doi = 10.2307/2007348
 | mr = 0669663
 | issue = 160
 | jstor = 2007348
 | ref = Zagier1982| doi-access = free
 }}</ref>  The conjecture was proved {{Disputed inline|Status of the asymptotic formula|date=July 2016}} by [Greg McShane](/source/Greg_McShane) and [Igor Rivin](/source/Igor_Rivin) in 1995 using techniques from [hyperbolic geometry](/source/hyperbolic_geometry).<ref>{{cite journal
 | author1 = Greg McShane
 | author2 = Igor Rivin
 | title = Simple curves on hyperbolic tori
 | journal = Comptes Rendus de l'Académie des Sciences, Série I
 | volume = 320
 | year = 1995
 | number = 12
 | ref = McShane1995}}</ref>

The ''n''th [Lagrange number](/source/Lagrange_number) can be calculated from the ''n''th Markov number with the formula

:<math>L_n = \sqrt{9 - {4 \over {m_n}^2}}.\,</math>

The Markov numbers are sums of (non-unique) pairs of squares.

==Markov's theorem==

{{Distinguish|Markov theorem}}

{{harvs|txt|last=Markoff|year1=1879|year2=1880}} showed that if

:<math>f(x,y) = ax^2+bxy+cy^2</math>

is an [indefinite](/source/indefinite_quadratic_form) [binary quadratic form](/source/binary_quadratic_form) with [real](/source/real_number) coefficients and [discriminant](/source/discriminant_of_a_quadratic_form) <math>D = b^2-4ac</math>, then there are integers ''x'',&nbsp;''y'' for which ''f'' takes a nonzero value of [absolute value](/source/absolute_value) at most

:<math>\frac{\sqrt D}{3}</math>

unless ''f'' is a ''Markov form'':<ref>Cassels (1957) p.39</ref> a constant times a form
:<math>px^2+(3p-2a)xy+(b-3a)y^2</math>
such that
:<math>\begin{cases} 0<a<p/2,\\
aq\equiv\pm r\pmod p,\\
 bp-a^2=1,
\end{cases}</math>
where (''p'',&nbsp;''q'',&nbsp;''r'') is a Markov triple.

==Matrices==
Let tr denote the [trace](/source/trace_(linear_algebra)) function over [matrices](/source/matrix_(mathematics)). If ''X'' and ''Y'' are in [SL](/source/special_linear_group)<sub>2</sub>('''[<math>\mathbb{C}</math>](/source/complex_number)'''), then

: <math> \operatorname{tr}(X) \operatorname{tr}(Y) \operatorname{tr}(XY) + \operatorname{tr}(XYX^{-1}Y^{-1}) + 2 = \operatorname{tr}(X)^2 + \operatorname{tr}(Y)^2 + \operatorname{tr}(XY)^2 </math>

so that if <math display=inline> \operatorname{tr}(XYX^{-1}Y^{-1}) = -2 </math> then

: <math> \operatorname{tr}(X) \operatorname{tr}(Y) \operatorname{tr}(XY) = \operatorname{tr}(X)^2 + \operatorname{tr}(Y)^2 + \operatorname{tr}(XY)^2 </math>

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 ''X''⋅''Y''⋅''Z''&nbsp;=&nbsp;[I](/source/identity_matrix) then tr(''XtY'')&nbsp;=&nbsp;tr(''Z''), so more symmetrically if ''X'', ''Y'', and ''Z'' are in SL<sub>2</sub>([<math>\mathbb{Z}</math>](/source/integer)) with ''X''⋅''Y''⋅''Z''&nbsp;=&nbsp;I and the [commutator](/source/Commutator) of two of them has trace −2, then their traces/3 are a Markov triple.<ref>Aigner (2013) Chapter 4, "The Cohn Tree", pp. 63–77</ref>

==See also==
*[Markov spectrum](/source/Markov_spectrum)

== Notes ==
<references/>

==References==
* {{cite book | last=Aigner | first=Martin | author-link=Martin Aigner | title=Markov's Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings | publisher=Springer | publication-place=Cham Heidelberg | date=2013-07-29 | isbn=978-3-319-00887-5 | mr=3098784}}
* {{cite book | first=J.W.S. | last=Cassels | author-link=J. W. S. Cassels | title=An Introduction to Diophantine Approximation | series=Cambridge Tracts in Mathematics and Mathematical Physics | volume=45 | publisher=[Cambridge University Press](/source/Cambridge_University_Press) | year=1957 | zbl=0077.04801 }}
* {{cite book | first1=Thomas | last1=Cusick | first2=Mary | last2=Flahive | title=The Markoff and Lagrange Spectra | series=Math. Surveys and Monographs | volume=30 | publisher=[American Mathematical Society](/source/American_Mathematical_Society) | location=Providence, RI | year=1989 | isbn=0-8218-1531-8 | zbl=0685.10023 }}
* {{cite book|first=Richard K. | last=Guy | author-link=Richard K. Guy| title=Unsolved Problems in Number Theory| publisher=[Springer-Verlag](/source/Springer-Verlag)| year=2004|isbn=0-387-20860-7| zbl=1058.11001 | pages=263–265 }}
* {{eom|id=m/m062540|first=A.V.|last= Malyshev|title=Markov spectrum problem}}
* {{Cite journal | last1=Markoff | first1=A. | author-link = Andrey Markov | title=Sur les formes quadratiques binaires indéfinies | journal=[Mathematische Annalen](/source/Mathematische_Annalen) | issn=0025-5831 }}
:: {{cite journal | last1=Markoff | first1=A. | author-link = Andrey Markov|title=First memoir| journal=[Mathematische Annalen](/source/Mathematische_Annalen) | year=1879 | doi=10.1007/BF02086269 | volume=15 | pages=381–406 | issue=3–4 | s2cid=179177894 |url=https://gdz.sub.uni-goettingen.de/id/PPN235181684_0015?tify=%7B%22view%22:%22info%22,%22pages%22:%5B393%5D%7D}}<!--- ref=Markoff1879--->
:: {{cite journal | last1=Markoff | first1=A. | author-link = Andrey Markov|title=Second memoir| journal=[Mathematische Annalen](/source/Mathematische_Annalen) | year=1880 | doi=10.1007/BF01446234 | volume=17 | pages=379–399 | issue=3 | s2cid=121616054 |url=https://gdz.sub.uni-goettingen.de/id/PPN235181684_0017?tify=%7B%22view%22:%22info%22,%22pages%22:%5B394%5D%7D}}<!--- ref=Markoff1880--->

Category:Diophantine equations
Category:Diophantine approximation
Category:Fibonacci numbers

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Adapted from the Wikipedia article [Markov number](https://en.wikipedia.org/wiki/Markov_number) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Markov_number?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.