Markov number

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Short description: Solution to x² + y² + z² = 3xyz

Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equation

[math]\displaystyle{ x^2 + y^2 + z^2 = 3xyz,\, }[/math]

studied by Andrey Markoff (1879, 1880).

The first few Markov numbers are

1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325, ... (sequence A002559 in the OEIS)

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

The first levels of the Markov number tree

There are two simple ways to obtain a new Markov triple from an old one (xyz). 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 (xyz) is a Markov triple then so is (xy, 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 is connected; 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 three neighbors (5, 13, 194), (1, 13, 34) and (1, 2, 5) in the Markov tree if z 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.

All the Markov numbers on the regions adjacent to 2's region are odd-indexed Pell numbers (or numbers n such that 2n2 − 1 is a square, OEISA001653), and all the Markov numbers on the regions adjacent to 1's region are odd-indexed Fibonacci numbers (OEISA001519). Thus, there are infinitely many Markov triples of the form

[math]\displaystyle{ (1, F_{2n-1}, F_{2n+1}),\, }[/math]

where Fk is the kth Fibonacci number. Likewise, there are infinitely many Markov triples of the form

[math]\displaystyle{ (2, P_{2n-1}, P_{2n+1}),\, }[/math]

where Pk is the kth Pell number.[2]

Other properties

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

The unicity conjecture, as remarked by Frobenius in 1913,[4] 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.[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]

Odd Markov numbers are 1 more than multiples of 4, while even Markov numbers are 2 more than multiples of 32.[9]

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

[math]\displaystyle{ m_n = \tfrac13 e^{C\sqrt{n}+o(1)} \quad\text{with } C = 2.3523414972 \ldots\,. }[/math]

The error [math]\displaystyle{ (\log(3m_n)/C)^2 - n }[/math] is plotted below.

Error in the approximation of large Markov numbers

Moreover, he pointed out that [math]\displaystyle{ x^2 + y^2 + z^2 = 3xyz + 4/9 }[/math], an approximation of the original Diophantine equation, is equivalent to [math]\displaystyle{ f(x)+f(y)=f(z) }[/math] with f(t) = arcosh(3t/2).[10] The conjecture was proved[disputed ] by Greg McShane and Igor Rivin in 1995 using techniques from hyperbolic geometry.[11]

The nth Lagrange number can be calculated from the nth Markov number with the formula

[math]\displaystyle{ L_n = \sqrt{9 - {4 \over {m_n}^2}}.\, }[/math]

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

Markov's theorem

Markoff (1879, 1880) showed that if

[math]\displaystyle{ f(x,y) = ax^2+bxy+cy^2 }[/math]

is an indefinite binary quadratic form with real coefficients and discriminant [math]\displaystyle{ D = b^2-4ac }[/math], then there are integers xy for which f takes a nonzero value of absolute value at most

[math]\displaystyle{ \frac{\sqrt D}{3} }[/math]

unless f is a Markov form:[12] a constant times a form

[math]\displaystyle{ px^2+(3p-2a)xy+(b-3a)y^2 }[/math]

such that

[math]\displaystyle{ \begin{cases} 0\lt a\lt p/2,\\ aq\equiv\pm r\pmod p,\\ bp-a^2=1, \end{cases} }[/math]

where (pqr) is a Markov triple.

Matrices

Let tr denote the trace function over matrices. If X and Y are in SL2([math]\displaystyle{ \mathbb{C} }[/math]), then

[math]\displaystyle{ \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]\displaystyle{ \operatorname{tr}(XYX^{-1}Y^{-1}) = -2 }[/math] then

[math]\displaystyle{ \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 XYZ = I then tr(XtY) = tr(Z), so more symmetrically if X, Y, and Z are in SL2([math]\displaystyle{ \mathbb{Z} }[/math]) with XYZ = I and the commutator of two of them has trace −2, then their traces/3 are a Markov triple.[13]

See also

Notes

  1. Cassels (1957) p.28
  2. OEISA030452 lists Markov numbers that appear in solutions where one of the other two terms is 5.
  3. Cassels (1957) p.27
  4. Frobenius, G. (1913). "Über die Markoffschen Zahlen". S. B. Preuss Akad. Wiss.: 458–487. 
  5. Guy (2004) p.263
  6. 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. 
  7. Rabideau, Michelle; Schiffler, Ralf (2020). "Continued fractions and orderings on the Markov numbers". Advances in Mathematics 370: 107231. doi:10.1016/j.aim.2020.107231. 
  8. Lee, Kyungyong; Li, Li; Rabideau, Michelle; Schiffler, Ralf (2023). "On the ordering of the Markov numbers". Advances in Applied Mathematics 143: 102453. doi:10.1016/j.aam.2022.102453. 
  9. Zhang, Ying (2007). "Congruence and Uniqueness of Certain Markov Numbers". Acta Arithmetica 128 (3): 295–301. doi:10.4064/aa128-3-7. Bibcode2007AcAri.128..295Z. http://journals.impan.gov.pl/aa/Inf/128-3-7.html. 
  10. Zagier, Don B. (1982). "On the Number of Markoff Numbers Below a Given Bound". Mathematics of Computation 160 (160): 709–723. doi:10.2307/2007348. 
  11. Greg McShane; Igor Rivin (1995). "Simple curves on hyperbolic tori". Comptes Rendus de l'Académie des Sciences, Série I 320 (12). 
  12. Cassels (1957) p.39
  13. Aigner, Martin (2013), "The Cohn tree", Markov's Theorem and 100 Years of the Uniqueness Conjecture, Springer, pp. 63–77, doi:10.1007/978-3-319-00888-2_4, ISBN 978-3-319-00887-5 .

References

Markoff, A. (1879). "First memoir". Mathematische Annalen 15 (3–4): 381–406. doi:10.1007/BF02086269. https://gdz.sub.uni-goettingen.de/id/PPN235181684_0015?tify=%7B%22view%22:%22info%22,%22pages%22:%5B393%5D%7D. 
Markoff, A. (1880). "Second memoir". Mathematische Annalen 17 (3): 379–399. doi:10.1007/BF01446234. https://gdz.sub.uni-goettingen.de/id/PPN235181684_0017?tify=%7B%22view%22:%22info%22,%22pages%22:%5B394%5D%7D.