Markov number

A Markoff number (after Andrei Markov ) is a natural number x, y, or z, as the solution of the diophantine equation Markoff

Occurs. The first Markoff numbers

They are parts of the solutions of the Markov equation, of which the first (1, 1, 1 ), (1, 1, 2 ), (1, 2, 5 ), (1, 5, 13 ), ( 2, 5, 29) loud. The solutions are also referred to as Markov triplet.

Markoff numbers come in the theory of quadratic forms and diophantine approximations ago: If m is a Markoff number, is both an element of the so-called Markoff spectrum ( square shapes ) and the Lagrange spectrum ( Diophantine approximations ).

Properties

There are an infinite number of Markoff numbers and triples. Since the Markov equation is symmetric in the variables, you can see the Lösungstripel (x, y, z) the size ordered by x ≤ y ≤ z specify. With the exception of the two smaller triple ( 1,1,1 ) and ( 1,1,2 ) are the Lösungstripel (x, y, z) in three different speeds. A long time examined - but still unproven - assumption states that the largest element z of a triplet already determined the Markoff triple ( x, y, z).

The Markov numbers can be arranged as shown at right in a tree. The adjacent to Region 1 Markoff numbers are the Fibonacci numbers with odd i the adjacent to the Region 2 Markoff numbers are the so-called Pell numbers with odd i

Is a Markoff number m is odd, then it satisfies the congruence m ≡ 1 mod 4 and if it is even, then we have m ≡ 2 mod 32 The three Markoff numbers of a triple are always pairwise relatively prime.

The generation of new Markoff triples from known

Can be obtained from a solution ( x, y, z) of the equation by means of Markov (x, y, z) → (x, y, 3xy - z) produce other solutions. It is not necessary that the solution with which one starts is sorted by size. The different arrays of x, y and z may have different triple ( x, y, 3xy - z) produced.

It is assumed, for example, (1, 5, 13), then one gets the three adjacent triple ( 5, 13, 194 ), (1, 13, 34) and (1, 2, 5) in the Markov tree when x equal to 1, 5 or 13 is. Turning to (x, y, z) → (x, y, 3xy - z) at twice without re-sort the entries in the triple, so you get the Ausgangstripel again.

Starting with (1, 1, 2) and swapped perpetually y and z before each transformation, we generated so that the above-mentioned Markoff triples containing Fibonacci numbers. With the same Starttripel but with inversion of x and z is produced Pell solutions.

What is the nth Markov number?

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

( here the O-notation is described by E. Landau used). The error is illustrated in the adjacent figure. The 1000th Markoff number is about 6 · 1031.