Markov number

 ( Diophantine Equations ) 

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

$x^2\; +\; y^2\; +\; z^2\; =\; 3xyz,\backslash ,$

studied by .

The first few Markov numbers are

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

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.

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Markov tree There are two simple ways to obtain a new Markov triple from an old one ( x,  y,  z). First, one may permutation 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 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 by a sequence of these operations.Cassels (1957) p.28 If one starts, as an example, with we get its three neighbors , and in the Markov tree if z is set to 1, 5 and 13, respectively. For instance, starting with and trading y and z before each iteration of the transform lists Markov triples with . Starting with that same triplet and trading x and z before each iteration gives the triples with .

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

$(1,\; F\_\{2n-1\},\; F\_\{2n+1\}),\backslash ,$

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

$(2,\; P\_\{2n-1\},\; P\_\{2n+1\}),\backslash ,$

where P _k is the kth Pell number. lists Markov numbers that appear in solutions where one of the other two terms is 5.

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Other properties Aside from the two smallest singular triples (1, 1, 1) and (1, 1, 2), every Markov triple consists of three distinct integers.Cassels (1957) p.27

The unicity conjecture, as remarked by Frobenius in 1913, 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.Guy (2004) p.263 Martin AignerAigner (2013) examines several weaker variants of the unicity conjecture. His fixed numerator conjecture was proved by Rabideau and Schiffler in 2020, while the fixed denominator conjecture and fixed sum conjecture were proved by Lee, Li, Rabideau and Schiffler in 2023.

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.Aigner (2013) p. 55 Furthermore, if $m$ is a Markov number then none of the prime divisors of $9m^2-4$ is congruent to 3 modulo 4. An even Markov number is 2 more than a multiple of 32.

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

$m\_n\; =\; \backslash tfrac13\; e^\{C\backslash sqrt\{n+o(1)\}\}\; \backslash quad\backslash text\{with\; \}\; C\; =\; 2.3523414972\; \backslash ldots\backslash ,.$

The error $o(1)\; =\; (\backslash log(3m\_n)/C)^2\; -\; n$ is plotted below.

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

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

$L\_n\; =\; \backslash sqrt\{9\; -\; \{4\; \backslash over\; \{m\_n\}^2\}\}.\backslash ,$

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

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Markov's theorem

showed that if
     

$f(x,y)\; =\; ax^2+bxy+cy^2$

is an indefinite binary quadratic form with real number coefficients and discriminant $D\; =\; b^2-4ac$, then there are integers x,  y for which f takes a nonzero value of absolute value at most

$\backslash frac\{\backslash sqrt\; D\}\{3\}$

unless f is a Markov form:Cassels (1957) p.39 a constant times a form

$px^2+(3p-2a)xy+(b-3a)y^2$

such that

aq\equiv\pm r\pmod p,\\

bp-a^2=1,
     

\end{cases} where ( p,  q,  r) is a Markov triple.

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Matrices Let tr denote the trace function over matrices. If X and Y are in SL₂( complex number), then

$\backslash operatorname\{tr\}(X)\; \backslash operatorname\{tr\}(Y)\; \backslash operatorname\{tr\}(XY)\; +\; \backslash operatorname\{tr\}(XYX^\{-1\}Y^\{-1\})\; +\; 2\; =\; \backslash operatorname\{tr\}(X)^2\; +\; \backslash operatorname\{tr\}(Y)^2\; +\; \backslash operatorname\{tr\}(XY)^2$

so that if $\backslash operatorname\{tr\}(XYX^\{-1\}Y^\{-1\})\; =\; -2$ then

$\backslash operatorname\{tr\}(X)\; \backslash operatorname\{tr\}(Y)\; \backslash operatorname\{tr\}(XY)\; =\; \backslash operatorname\{tr\}(X)^2\; +\; \backslash operatorname\{tr\}(Y)^2\; +\; \backslash operatorname\{tr\}(XY)^2$

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 = identity matrix then tr( XtY) = tr( Z), so more symmetrically if X, Y, and Z are in SL₂(integer) with X⋅ Y⋅ Z = I and the commutator of two of them has trace −2, then their traces/3 are a Markov triple.Aigner (2013) Chapter 4, "The Cohn Tree", pp. 63–77

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See also

• Markov spectrum

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Notes

• Martin Aigner (2013). 9783319008875, Springer. ISBN 9783319008875
• (1989). 9780821815311, American Mathematical Society. ISBN 9780821815311
• Richard K. Guy (2025). 9780387208602, Springer-Verlag. ISBN 9780387208602

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Categories: Diophantine Equations, Diophantine Approximation, Fibonacci Numbers

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