Markov Numbers and the Uniqueness Conjecture

A guided research project in Geometry & Dynamics.

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Question

Does a Markov number determine the Markov triple in which it is largest?

Mutations create an infinite tree of solutions in which the uniqueness pattern can be tested.
Mutations create an infinite tree of solutions in which the uniqueness pattern can be tested.

Goals

  • Generate Markov triples and study the pattern of their largest entries.
  • Connect the number-theoretic question with traces of closed geodesics.

Method

Explore the Markov equation through its mutation tree and its geometric meaning.

  • Start from \((1,1,1)\) and generate new triples by Markov mutations.
  • Record repeated largest values and organize the triples computationally.
  • Study how the same data appears in matrices, traces, or simple geodesics.