Markoff number

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The first entries in the tree of Markoff numbers

A Markoff number (after Andrei Andrejewitsch Markow ) is a natural number or , as a solution of the Diophantine Markoff equation

occurs. The first Markoff numbers are

1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325, ...

They are parts of the solutions to Markoff's equation, the first of which are. The solutions are also referred to as Markoff triples .

Markoff numbers occur in the theory of quadratic forms and Diophantine approximations : If a Markoff number is an element of the so-called Markoff spectrum (quadratic forms) as well as the Lagrange spectrum (Diophantine approximations).


There are an infinite number of Markoff numbers and triples. Since the Markoff equation is symmetrical in the variables, the solution triples can also be given in order of size . With the exception of the two smallest triples and , the solution triples consist of three different numbers. A conjecture that has been investigated for a long time - but has not yet been proven - states that the largest element of a triple already determines the Markoff triple .

The Markoff numbers can be arranged in a tree as shown on the right. The Markoff numbers neighboring region 1 are the Fibonacci numbers with odd . The Markoff numbers adjacent to region 2 are the so-called Pell numbers with odd .

If a Markoff number is odd, then it fulfills the congruence and if it is even then it holds . The three Markoff numbers of a triple are always in pairs relatively prime .

The generation of new Markoff triples from known

One can generate further solutions from one solution of the Markoff equation . It is not necessary that the solution you start with is ordered by size. The different arrangements of and can produce different triples .

If you take for example , then you get the three neighboring triples and in the Markoff tree if you place or bet. If you apply twice without rearranging the entries in the triple, you get the starting triple again.

Error in approximation for the first 1000 Markoff numbers

If you start with and swap continuously and before each transformation, you generate the Markoff triples mentioned above, which contain Fibonacci numbers. The Pell solutions are generated with the same starting triple but with swapping and .

What is the nth Markoff number?

In 1982 Don Zagier proved an asymptotic formula for the number of Markoff triples below a bound and hypothesized that the -th Markoff number is asymptotically given by


( E. Landau's O notation is used here). The error is illustrated in the adjacent figure. The 1000th Markoff number is approx .


  • Thomas Cusick, Mari Flahive: The Markoff and Lagrange spectra . In: Math. Surveys and Monographs , 30, 1989, AMS, Providence
  • Serge Perrine: La théorie de Markoff et ses développements . Tessier & Ashpool, 2002, arxiv : math-ph / 0307032
  • Caroline Series : The Geometry of Markoff Numbers . In: The Mathematical Intelligencer , 7 (3), 1985, pp. 20-29.
  • Eric W. Weisstein : Markov number . In: MathWorld (English).

Individual evidence

  1. See also the section “The Markoff Numbers” in Paulo Ribenboim's book My Numbers, My Friends : Google Books
  2. The Markoff numbers are the sequence A002559 in Neil Sloane's Online Encyclopedia of Integer Sequences .
  3. Norbert Riedel's solution approach from 2007 ( Markoff Equation and Nilpotent Matrices , arxiv : 0709.1499 ) is discussed in the long article by Serge Perrine: De Frobenius à Riedel: analyze des solutions de l'équation de Markoff , Archive-Ouvertes ( PDF; 713 kB).
  4. These are sufficient, with the start values and , the recursion . The Odd Pell numbers have the property that is a square number (they are solutions of Pell's equation ).
  5. Ying Zhang: Congruence and Uniqueness of Certain Markov Numbers , Acta Arithmetica 128 3, 2007, 295-301
  6. It is true .
  7. ^ Don B. Zagier: On the Number of Markoff Numbers Below a Given Bound . In: Mathematics of Computation , 160, 1982, pp. 709–723, (PDF; 1.2 MB)
  8. See the lecture by M. Waldschmidt ( Memento of the original from February 24, 2014 in the Internet Archive ; PDF; 4.2 MB) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot /
  9. Follow A002559 in OEIS