Markov number

A positive integer that occurs in a positive-integer solution of the equation x² + y² + z² = 3xyz. The equation’s solutions form a tree generated by Vieta mutations.

Markov triple: A triple of positive integers satisfying x² + y² + z² = 3xyz. Each Markov number is one coordinate in at least one such solution.
Andrey Markov: A Russian mathematician known for work in probability, number theory, and continued fractions. His study of best rational approximations led to the equation defining these numbers.
Once-punctured torus: A torus with one point removed. Simple closed curves on this surface are labeled by Markov triples through their intersection geometry.
Markov uniqueness conjecture: The conjecture that each Markov number occurs in exactly one Markov triple, up to permutation. It asks whether a number can occupy different positions in distinct solution triples.
Vieta jumping: A descent method for Diophantine equations that replaces one root of a quadratic with the other. It turns one Markov triple into a neighboring triple while preserving the equation.
Lagrange spectrum: A set of constants measuring how well irrational numbers can be approximated by rationals. Markov’s equation describes values at the beginning of this spectrum.
Teichmüller space: The space of marked conformal or hyperbolic structures on a surface. Markov triples appear in coordinates describing structures on the punctured torus.
Markov spectrum: The spectrum of constants governing the quality of rational approximation to irrational numbers. The broader spectrum raises questions about where Markov values lie and how they accumulate.
Markov tree: The tree whose vertices are Markov triples and whose edges connect triples related by Vieta mutation. Its vertices organize how the equation generates Markov numbers.
Continued fraction: A representation of a number as an integer plus a sequence of reciprocal terms. Continued fractions supplied the approximation problem behind Markov’s discoveries.

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