Math Circles
Mathematics Department • Xavier University of Louisiana
Spring 2026
Week 1 Saturday • February 21, 2026

Markov Numbers & Markov Triples

Notes + exercises
Diophantine equation → cluster ideas
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Mathematics Department • Xavier University of Louisiana
Math Circles — Spring 2026
Week 1 — Saturday February 21

1) Markov triples (M.T.)

A triple of positive integers \((x,y,z)\) is called a Markov triple if it satisfies the Markov equation

\[ x^2 + y^2 + z^2 \;=\; 3xyz. \]

Any integer appearing in a Markov triple is called a Markov number.

Examples

  • \((1,1,1)\) since \(1^2+1^2+1^2=3=3\cdot1\cdot1\cdot1\).
  • \((1,1,2)\) since \(1^2+1^2+2^2=6=3\cdot1\cdot1\cdot2\).

Exercise

Check that the following are Markov triples:

\[ (1,2,5),\quad (1,5,13),\quad (2,5,29),\quad (1,13,34). \]

Historical note. Markov triples were studied by A. Markoff (late 1800s). In 1913, Frobenius conjectured a “uniqueness” property: roughly, that the largest entry in a Markov triple determines the other two (up to permutation). The full conjecture remains open, with partial progress via cluster-algebra methods.

2) Properties of Markov triples

Two natural questions:

  • Can we obtain a new Markov triple from a given one?
  • How many Markov triples are there?

Move (1): permutation

If \((x,y,z)\) is a Markov triple, then any rearrangement is also a Markov triple (symmetry).

Example. From \((1,2,5)\), we get

\[ (2,1,5),\ (2,5,1),\ (1,5,2),\ (5,1,2),\ (5,2,1). \]

Move (2): a Vieta-type involution

Start from the Markov equation and isolate the \(z\)-dependence:

\[ x^2 + y^2 + z^2 - 3xyz = 0 \qquad\Longleftrightarrow\qquad x^2 + y^2 + z\,(z-3xy)=0. \]

Define a new integer

\[ z' \;=\; 3xy - z. \]

Then \((x,y,z')\) is also a Markov triple (a “Vieta move”).

Exercise

Using Moves (1) and (2), build a chain of Markov triples starting from \((1,1,1)\). (Optional: look for patterns related to Fibonacci numbers.)

3) Mutation of a quiver

A quiver is a directed graph (vertices + arrows). In this context we exclude:

  • loops (an arrow from a vertex to itself),
  • 2-cycles (a pair of opposite arrows between two vertices).

Let \(Q\) be a quiver with vertices \(1,2,\dots,n\). Mutation at a vertex \(k\), denoted \(\mu_k(Q)\), is defined by:

  1. If there is a path \(i \to k \to j\), add an arrow \(i \to j\).
  2. Reverse all arrows incident to \(k\).
  3. Delete any 2-cycles created by the previous steps.

This mutation process is central in cluster algebra theory; in modern treatments, Markov triples can be studied via repeated mutations of a 3-vertex quiver.