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).

There are two simple ways to obtain a new Markov triple from an old one (x, y, z).

Applying this operation twice returns the same triple one started with.

[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.

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

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

[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.

[10] In his 1982 paper, Don Zagier conjectured that the nth Markov number is asymptotically given by The error

, an approximation of the original Diophantine equation, is equivalent to

[11] The conjecture was proved [disputed – discuss] by Greg McShane and Igor Rivin in 1995 using techniques from hyperbolic geometry.

Markoff (1879, 1880) showed that if is an indefinite binary quadratic form with real coefficients and discriminant

Let tr denote the trace function over matrices.