Markov number

A Markov numberMathworldPlanetmath is an integer x, y or z that fits in the Diophantine equationMathworldPlanetmath


and gives a Lagrange number


(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 numbersDlmfMathworldPlanetmath with odd index, and the numbers on 2’s branch are Pell numbersMathworldPlanetmath 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 x1mod4 (or y or z) and an even Markov number x2mod8. Ying Zhang used this to prove that even Markov numbers satisfy the sharper congruenceMathworldPlanetmathPlanetmath x2mod32, which he calls the best possible since the first two even Markov numbers are 2 and 34.


  • 1 Ying Zhang, “Congruence and Uniqueness of Certain Markov Numbers” Acta Arithmetica 128 3 (2007): 297
Title Markov number
Canonical name MarkovNumber
Date of creation 2013-03-22 15:46:19
Last modified on 2013-03-22 15:46:19
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 10
Author CompositeFan (12809)
Entry type Definition
Classification msc 11D72
Classification msc 11J06
Synonym Markoff number