A Markov triple is a positive integer solution (x,y,z) of the Markov equation
A Markov number is any part x,y, or z of a Markov triple.
There are infinitely many Markov triples, and all of them may be obtained from (1,1,1) by a simple recurrent procedure. In particular, every second Fibonacci number is also a Markov numbers.
There are many puzzles around the Markov equation. For example, there is a Uniqueness Conjecture (formulated by Frobenius in 1913) which says that if (x,y,z) and (x',y',z') are two Markov triples with minimal elements x=x'≠ 1, then the triples do coincide. The conjecture is still neither proved nor disproved (even with extensive use of computers!).
One can study Markov numbers in may different ways, and there are tight connections to several fields in mathematics such as
- number theory,
- continued fractions,
- hyperbolic geometry,
- group theory,
- cluster algebras.