# 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

 $L_{x}=\sqrt{9-{4\over x^{2}}}$

(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 $x\equiv 1\mod 4$ (or $y$ or $z$) and an even Markov number $x\equiv 2\mod 8$. Ying Zhang used this to prove that even Markov numbers satisfy the sharper congruence $x\equiv 2\mod 32$, which he calls the best possible since the first two even Markov numbers are 2 and 34.

## References

• 1 Ying Zhang, “Congruence and Uniqueness of Certain Markov Numbers” Acta Arithmetica 128 3 (2007): 297
Title Markov number MarkovNumber 2013-03-22 15:46:19 2013-03-22 15:46:19 CompositeFan (12809) CompositeFan (12809) 10 CompositeFan (12809) Definition msc 11D72 msc 11J06 Markoff number