Markov numbers

Some (mostly on-line) resources

__Number Theoretic properties__

- Don Zagier, On the Number of Markoff Numbers Below a Given Bound.
- Pietro Corvaja, Umberto Zannier, On the Greatest Prime Factor of Markov Pairs.

__Uniqueness conjecture:__

- Markov's Theorem and 100 Years of the Uniqueness Conjecture. A Mathematical Journey from Irrational Numbers to Perfect Matchings by Martin Aigner.
- D.Rosen, G.S.Patterson, Some Numerical Evidence Concerning the Uniqueness of the Markov Numbers.
- I. Borosh, More numberical evidence on the uniqueness of Markov numbers.
- K.R. Matthews, J.P. Robertson, An equivalent form of the unicity conjecture for Markoff numbers.
- Ying Zhang, Congruence and uniqueness of certain Markoff numbers .
- Ying Zhang, An elementary proof of uniqueness of Markoff numbers which are prime powers .
- Mong Lung Lang, Ser Peow Tan, A simple proof of the Markoff conjecture for prime powers.

__Geometry of Markov numbers:__

- Caroline Series, The Geometry of Markoff Numbers.
- Yi Huang, Notes for "The Geometry of Markoff Numbers” by Caroline Series.
- Yi Huang, Paul Norbury, Simple geodesics and Markoff quads.
- Caroline Series, Symbolic Dynamics for Geodesic Flows.
- A.F.Beardon, J.Lehner, M.Sheingorn, Closed geodesics on a riemann surface with application to the Markov spectrum.
- Greg McShane, Hugo Parlier, Simple closed geodesics of equal length on a torus .
- Andrew Haas, The geometry of Markoff forms.
- Hervey Cohn, Approach to Markoff's minimal forms through modular functions.

__Continued Fractions__

- Oleg Karpenkov, Geometry of Continued Fractions.
- Caroline Series, Continued Fractions and Hyperbolic Geometry.
- Wieb Bosma, Cor Kraaikamp (and 9 students), Continued Fractions, notes of a course.
- Enrico Bombieri, Continued fractions and the Markoff tree.
- Ian Short, The hyperbolic geometry of continued fractions.

__Farey and Euclid trees:__

- Farey sequences, the Farey map and the Farey Tree webpage by Matthew R. Watkins.
- Donald Joseph Mcginn, Generalized Markoff Equations, Euclid Trees, and Chebyshev Polynomials.
- Sophie Morier-Genoud, Valentin Ovsienko, Serge Tabachnikov, SL_2(Z)-tilings of the torus, Coxeter-Conway friezes and Farey triangualtions .
- Jonathan Ainsworth, Michael Dawson, John Pianta, James Warwick, The Farey sequence, Report on 4H project, Edinbourgh, 2012.

__Combinatorics of Markov numbers__

- Christophe Reutenauer, Christofel words and Markoff triples.
- James Propp, The combinatorics of frieze patterns and Markoff numbers.
- Andy Itsara, Gregg Musiker, James Propp, and Rui Viana, Combinatorial Interpretations for the Markov Numbers.

__Markov numbers and cluster algebras__

- Xueyuan Peng, Jie Zhang, Cluster algebras and Markoff numbers.
- Philipp Lampe, Diophantine equations via cluster transformations .
- Ilke Canakci, Ralf Schiffler, Cluster algebras and continued fractions.

__Introduction to cluster algebras:__

- Philipp Lampe, Cluster algebras.
- Lauren Williams, Cluster algebras: an introduction.
- Sergey Fomin, Lauren Williams, Andrei Zelevinsky, Introduction to Cluster Algebras, Chapters 1–3.

__Markov equation, generalised Markov equation:__

- Sara Maloni, Frederic Palesi, Ser Peow Tan, On the character variety of the hour-holed sphere.
- Andre Beineke, Thomas Brustle, Lutz Hille, Cluster-Cyclic Quivers with three Vertices and the Markov Equation, with an appendix by Otto Kerner .

__Some webpages:__

- Markov numbers by Tom Ace.
- Markov numbers, Wikipedia

__Slides of some talks:__

- Markov numbers by Kha H. Man.