Markov number

A Markov number is an integer $x$, $y$ or $z$ that fits in the Diophantine equation

$$x^2+y^2+z^2=3xyz$$

and gives a Lagrange number

$$L_x=\sqrt{9-\frac{4}{x^2}}$$

(or $y$ or $z$ as the case may be).

The solutions, (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), etc., can be put in a binary graph tree. Thus arranged, the numbers on 1’s branch are Fibonacci numbers with odd index, and the numbers on 2’s branch are Pell numbers with odd index.

Georg Frobenius proved that, with the exception of the smallest Markov triple, the numbers in a Markov triple are pairwise coprime. He also proved that an odd Markov number $x\equiv 1mod4$ (or $y$ or $z$) and an even Markov number $x\equiv 2mod8$. Ying Zhang used this to prove that even Markov numbers satisfy the sharper congruence $x\equiv 2mod32$, which he calls the best possible since the first two even Markov numbers are 2 and 34.