# Markov number

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

studied by Andrey Markoff (1879, 1880).

The first few Markov numbers are

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), etc.

There are infinitely many Markov numbers and Markov triples.

## Contents

## Markov tree[edit]

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.^{[1]} 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 *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 2*n*^{2} − 1 is a square, 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

where *F*_{x} is the *x*th Fibonacci number. Likewise, there are infinitely many Markov triples of the form

where *P*_{x} is the *x*th Pell number.^{[2]}

## Other properties[edit]

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* 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.^{[4]}

Odd Markov numbers are 1 more than multiples of 4, while even Markov numbers are 2 more than multiples of 32.^{[5]}

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

Moreover, he pointed out that , an approximation of the original Diophantine equation, is equivalent to with *f*(*t*) = arcosh(3*t*/2).^{[6]} The conjecture was proved^{[disputed – discuss]} by Greg McShane and Igor Rivin in 1995 using techniques from hyperbolic geometry.^{[7]}

The *n*th Lagrange number can be calculated from the *n*th Markov number with the formula

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

## Markov's theorem[edit]

Markoff (1879, 1880) showed that if

is an indefinite binary quadratic form with real coefficients and discriminant , then there are integers *x*, *y* for which *f* takes a nonzero value of absolute value at most

unless *f* is a *Markov form*:^{[8]} a constant times a form

such that

where (*p*, *q*, *r*) is a Markov triple.

There is also a Markov theorem in topology, named after the son of Andrey Markov, Andrei Andreevich Markov.^{[9]}

## Matrices[edit]

Let Tr denote the trace function over matrices. If *X* and *Y* are in SL_{2}(**ℂ**), then

so that if Tr(*X*⋅*Y*⋅*X*^{−1} ⋅ *Y*^{−1}) = −2 then

- Tr(
*X*) Tr(*Y*) Tr(*X*⋅*Y*) = Tr(*X*)^{2}+ Tr(*Y*)^{2}+ Tr(*X*⋅*Y*)^{2}

In particular if *X* and *Y* also have integer entries then Tr(*X*)/3, Tr(*Y*)/3, and Tr(*X*⋅*Y*)/3 are a Markov triple. If *X*⋅*Y*⋅*Z* = 1 then Tr(*X*⋅*Y*) = Tr(*Z*), so more symmetrically if *X*, *Y*, and *Z* are in SL_{2}(ℤ) with *X*⋅*Y*⋅*Z* = 1 and the commutator of two of them has trace −2, then their traces/3 are a Markov triple.^{[10]}

## See also[edit]

## Notes[edit]

**^**Cassels (1957) p.28**^**OEIS: A030452 lists Markov numbers that appear in solutions where one of the other two terms is 5.**^**Cassels (1957) p.27**^**Guy (2004) p.263**^**Zhang, Ying (2007). "Congruence and Uniqueness of Certain Markov Numbers".*Acta Arithmetica*.**128**(3): 295–301. arXiv:math/0612620. Bibcode:2007AcAri.128..295Z. doi:10.4064/aa128-3-7. MR 2313995.**^**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. JSTOR 2007348. MR 0669663.**^**Greg McShane; Igor Rivin (1995). "Simple curves on hyperbolic tori".*Comptes Rendus de l'Académie des Sciences, Série I*.**320**(12).**^**Cassels (1957) p.39**^**Louis H. Kauffman,*Knots and Physics*, p. 95, ISBN 978-9814383011**^**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, MR 3098784.

## References[edit]

- Cassels, J.W.S. (1957).
*An introduction to Diophantine approximation*. Cambridge Tracts in Mathematics and Mathematical Physics.**45**. Cambridge University Press. Zbl 0077.04801. - Cusick, Thomas; Flahive, Mari (1989).
*The Markoff and Lagrange spectra*. Math. Surveys and Monographs.**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", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Markoff, A. "Sur les formes quadratiques binaires indéfinies".
*Mathematische Annalen*. Springer Berlin / Heidelberg. ISSN 0025-5831.

- Markoff, A. (1879). "First memory".
*Mathematische Annalen*.**15**(3–4): 381–406. doi:10.1007/BF02086269. - Markoff, A. (1880). "Second memory".
*Mathematische Annalen*.**17**(3): 379–399. doi:10.1007/BF01446234.

- Markoff, A. (1879). "First memory".