A Markov number orMarkoff number is a positiveintegerx,y orz that is part of a solution to the MarkovDiophantine equation

${\displaystyle x^2+y^2+z^2=3xyz,}$

studied byAndrey Markoff (1879,1880).

The first few Markov numbers are

1,2,5,13,29,34,89,169,194,233, 433, 610, 985, 1325, ... (sequenceA002559 in theOEIS)

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.

There are two simple ways to obtain a new Markov triple from an old one (x, y, z). First, one maypermute the 3 numbersx,y,z, so in particular one can normalize the triples so thatx ≤ y ≤ z. Second, if (x, y, z) is a Markov triple then so is (x, y, 3xy − 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 isconnected; in other words every Markov triple can be connected to(1,1,1) by a sequence of these operations.^[1] If one starts, as an example, with(1, 5, 13) we get its threeneighbors(5, 13, 194),(1, 13, 34) and(1, 2, 5) in the Markov tree ifz is set to 1, 5 and 13, respectively. For instance, starting with(1, 1, 2) and tradingy andz before each iteration of the transform lists Markov triples withFibonacci numbers. Starting with that same triplet and tradingx andz before each iteration gives the triples withPell numbers.

All the Markov numbers on the regions adjacent to 2's region areodd-indexed Pell numbers (or numbersn such that 2n² − 1 is asquare,OEIS: A001653), and all the Markov numbers on the regions adjacent to 1's region are odd-indexed Fibonacci numbers (OEIS: A001519). Thus, there are infinitely many Markov triples of the form

${\displaystyle (1,F_{2n-1},F_{2n+1}),}$

whereF_k is thekthFibonacci number. Likewise, there are infinitely many Markov triples of the form

${\displaystyle (2,P_{2n-1},P_{2n+1}),}$

whereP_k is thekthPell number.^[2]

Aside from the two smallestsingular triples (1, 1, 1) and (1, 1, 2), every Markov triple consists of three distinct integers.^[3]

Theunicity conjecture, as remarked byFrobenius in 1913,^[4] states that for a given Markov numberc, there is exactly one normalized solution havingc as its largest element:proofs of thisconjecture have been claimed but none seems to be correct.^[5]Martin Aigner^[6] examines several weaker variants of the unicity conjecture. His fixed numerator conjecture was proved by Rabideau and Schiffler in 2020,^[7] while the fixed denominator conjecture and fixed sum conjecture were proved by Lee, Li, Rabideau and Schiffler in 2023.^[8]

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.^[9] Furthermore, if${\displaystyle m}$ is a Markov number then none of the prime divisors of${\displaystyle 9m^2-4}$ is congruent to 3 modulo 4. Aneven Markov number is 2 more than a multiple of 32.^[10]

In his 1982 paper,Don Zagier conjectured that thenth Markov number is asymptotically given by

${\displaystyle m_n=\frac{1}{3}{e}^{C\sqrt{n+o(1)}}\text{with }C=2.3523414972\dots .}$

The error${\displaystyle o(1)=(\log (3m_n)/C{)}^2-n}$ is plotted below.

Moreover, he pointed out that${\displaystyle x^2+y^2+z^2=3xyz+4/9}$, an approximation of the original Diophantine equation, is equivalent to${\displaystyle f(x)+f(y)=f(z)}$ withf(t) =arcosh(3t/2).^[11] The conjecture was proved^{[disputed –discuss]} byGreg McShane andIgor Rivin in 1995 using techniques fromhyperbolic geometry.^[12]

ThenthLagrange number can be calculated from thenth Markov number with the formula

${\displaystyle L_n=\sqrt{9-\frac{4}{{m_n}^2}}.}$

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

Markoff (1879,1880) showed that if

${\displaystyle f(x,y)=a{x}^2+bxy+c{y}^2}$

is anindefinitebinary quadratic form withreal coefficients anddiscriminant${\displaystyle D=b^2-4ac}$, then there are integersx, y for whichf takes a nonzero value ofabsolute value at most

${\displaystyle \frac{\sqrt{D}}{3}}$

unlessf is aMarkov form:^[13] a constant times a form

${\displaystyle p{x}^2+(3p-2a)xy+(b-3a)y^2}$

such that

where (p, q, r) is a Markov triple.

Let tr denote thetrace function overmatrices. IfX andY are inSL₂(${\displaystyle \mathbb{C}}$), then

${\displaystyle \mathrm{tr}(X)\mathrm{tr}(Y)\mathrm{tr}(XY)+\mathrm{tr}(XY{X}^{-1}{Y}^{-1})+2=\mathrm{tr}(X{)}^2+\mathrm{tr}(Y{)}^2+\mathrm{tr}(XY{)}^2}$

so that if$\mathrm{tr}(XY{X}^{-1}{Y}^{-1})=-2$ then

${\displaystyle \mathrm{tr}(X)\mathrm{tr}(Y)\mathrm{tr}(XY)=\mathrm{tr}(X{)}^2+\mathrm{tr}(Y{)}^2+\mathrm{tr}(XY{)}^2}$

In particular ifX andY also have integer entries then tr(X)/3, tr(Y)/3, and tr(XY)/3 are a Markov triple. IfX⋅Y⋅Z = I then tr(XtY) = tr(Z), so more symmetrically ifX,Y, andZ are in SL₂(${\displaystyle \mathbb{Z}}$) withX⋅Y⋅Z = I and thecommutator of two of them has trace −2, then their traces/3 are a Markov triple.^[14]

• Markov spectrum

• Aigner, Martin (2013-07-29).Markov's Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings. Cham Heidelberg: Springer.ISBN 978-3-319-00887-5.MR 3098784.
• Cassels, J.W.S. (1957).An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 45.Cambridge University Press.Zbl 0077.04801.
• Cusick, Thomas; Flahive, Mary (1989).The Markoff and Lagrange spectra. Math. Surveys and Monographs. Vol. 30. Providence, RI:American Mathematical Society.ISBN 0-8218-1531-8.Zbl 0685.10023.
• Guy, Richard K. (2004).Unsolved Problems in Number Theory.Springer-Verlag. pp. 263–265.ISBN 0-387-20860-7.Zbl 1058.11001.
• Malyshev, A.V. (2001) [1994],"Markov spectrum problem",Encyclopedia of Mathematics,EMS Press
• Markoff, A. "Sur les formes quadratiques binaires indéfinies".Mathematische Annalen. Springer Berlin / Heidelberg.ISSN 0025-5831.

Markoff, A. (1879)."First memoir".Mathematische Annalen.15 (3–4):381–406.doi:10.1007/BF02086269.S2CID 179177894.

Markoff, A. (1880)."Second memoir".Mathematische Annalen.17 (3):379–399.doi:10.1007/BF01446234.S2CID 121616054.

• Diophantine equations
• Diophantine approximation
• Fibonacci numbers

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