Markov Triangles: Structure & Applications

• Markov triangles are sets of three positive integers satisfying the equation x² + R² + z² = 3xRz, forming the basis of an infinite ternary mutation tree.
• They exhibit structured recurrence relations and connections with Fibonacci, Pell, and Lucas sequences that organize the arithmetic progression of Markov numbers.
• Their unique properties extend to algebraic geometry, influencing lattice-point enumeration, classification of surface singularities, and birational invariants.

A Markov triangle (or Markov triplet) consists of three positive integers $\{x, R, z\}$ that satisfy the Markov Diophantine equation $x^2 + R^2 + z^2 = 3xRz$ (Gore, 4 Jun 2025, Urzúa et al., 2023, Fares, 20 Jan 2026). These solutions, called Markov numbers, form the vertices of an infinite trivalent graph known as the Markov tree. Markov triangles underlie deep arithmetic, geometric, and combinatorial phenomena, including recurrence relations, connections with Pell equations and birational geometry, explicit geometric constructions, and remarkable structures in lattice-point enumeration.

1. Diophantine Structure and Tree Construction

Markov triangles are precisely the ordered positive integer solutions $(x, R, z)$ to the equation $x^2 + R^2 + z^2 = 3xRz$, where $R > \max\{x, z\}$ is designated as the “region number.” Notable examples include the singular triplets $(1,1,1)$, $(1,2,1)$, and the first non-singular triplet $(1,5,2)$. Every non-singular triplet possesses three distinct children generated by “mutations,” forming an infinite ternary tree:

• Left move: $\{x, R, z\} \mapsto \{x, 3Rx - z, R\}$
• Right move: $\{x, R, z\} \mapsto \{R, 3Rz - x, z\}$
• Up move: Recovers the parent from a child

Iterating these mutations from any root constructs the full Markov tree, where each triplet appears once (Gore, 4 Jun 2025, Urzúa et al., 2023).

2. Regions, Branches, and Recurrence Relations

Triplet $\{x, R, z\}$ is said to “head” region $R$. The left and right edges of region $R$ are defined by repeated left or right mutations. The sequence of region numbers along each edge obeys a uniform second-order linear recurrence:

$a_{n+1} = 3R a_n - a_{n-1}$

Initial terms are specified by the first two edge entries. This recurrence organizes the arithmetic progression of Markov numbers and governs both Fibonacci and Pell branches. For the Fibonacci branch $$(1,1,1) \rightarrow (1,1,2) \rightarrow (1,2,5) \rightarrow (1,5,13) \rightarrow \dots$$, the Markov numbers along the spine are the odd-indexed Fibonacci numbers (Urzúa et al., 2023). The Pell branch $$(1,1,2) \rightarrow (1,2,5) \rightarrow (2,5,29) \rightarrow \dots$$ follows an analogous recurrence with different seeds.

Defining the Lucas $U$-sequence $U_k(P, Q)$ by $U_0=0$, $U_1=1$, $U_{k+1}=P U_k - Q U_{k-1}$ with $P=3R$, $Q=1$, closed-form expressions for edge region-numbers arise:

• Left edge: $L_k = z U_k(3R, 1) - x U_{k-1}(3R, 1)$
• Right edge: %%%%24%%%%

Generating functions for the edges are rational:

$$H_{gf}^L(n) = \frac{x - n z}{1 - 3R n + n^2}, \quad H_{gf}^R(n) = \frac{z - n x}{1 - 3R n + n^2}$$

Primitive examples include region $R=5$ with left-edge entries $1, 13, 194, 2897, \dots$ (Gore, 4 Jun 2025).

3. Pell Equations and Uniqueness of Markov Numbers

Each edge of the Markov tree corresponds to an integer solution to a specific Pell equation. Defining the associated Lucas $V$-sequence $V_0=2$, $V_1=3R$, $V_{k+1}=3R V_k - V_{k-1}$, one obtains:

$V_k^2 - D(R) U_k^2 = -(2R)^2,$

where $D(R) = (3R)^2 - 4$. This inhomogeneous Pell equation fully determines the edge sequence: all solutions are explicitly generated by powers of the fundamental unit $(3R + \sqrt{D})/2$ and a baseline seed. The Uniqueness Conjecture for Markov numbers is resolved: each such Pell equation admits exactly the region-numbers $U_k$ produced by the recurrence, so every Markov number $R$ is uniquely realized (Gore, 4 Jun 2025, Urzúa et al., 2023).

4. Special Squares and Closed-Form Formulae

Each Markov number $R$ can be expressed as the sum of two squares $R = \sigma_R^2 + \Lambda_R^2$. A distinguished algorithm selects “special square terms” $(\sigma_R, \Lambda_R)$ so that the smaller entries of the triplet $(x, R, z)$ are related via Brahmagupta–Fibonacci identities. The procedure is recursive: given $(x, R, z)$, one computes its sibling $s = 3xz - R$ and solves two linear equations for $(\sigma_R, \Lambda_R)$ using the special squares of $s$. Well-chosen initial values guarantee termination.

The sequence of special squares along either edge also satisfies the same recurrence kernel:

$$\sigma_{k+1} = 3R \sigma_k - \sigma_{k-1}, \quad \Lambda_{k+1} = 3R \Lambda_k - \Lambda_{k-1}$$

Generating functions for odd/even terms are explicit rational expressions:

$$K_{gf}^{odd}(n) = \frac{\alpha - \gamma n}{1 - 3R n + n^2}, \quad K_{gf}^{even}(n) = \frac{\beta - \delta n}{1 - 3R n + n^2}$$

Constants $\alpha, \beta, \gamma, \delta$ are read from the first two edge-triplets (Gore, 4 Jun 2025).

5. Lattice Geometry and Period Collapse

Markov triangles, constructed either via recursive integral-affine mutations or explicit coordinate models, have edge-length and angle-determinant data reflecting Markov triples (Fares, 20 Jan 2026). Any two entries of a Markov triple are coprime. A geometric mutation at a triangle vertex corresponds to an integral half-shear and translation, preserving the lattice-point count for all integer dilations.

The Ehrhart function $L_P(k)$, counting integer points in dilates of a convex polygon $P$, is in general a quasipolynomial of period dividing the denominator $D$. For Markov triangles, strong period collapse occurs: the period of $L_P$ always divides the largest Markov number $a$, regardless of the denominator. Irrational limits of Markov triangles (defined by Hausdorff convergence of rational sequences) retain this periodicity. For triangles placed in integral barycentric position, the period further collapses to $1$ (the Ehrhart function becomes a genuine polynomial).

Explicit formulae for $L_\Delta(k)$ use Fourier-Dedekind sums with periods strictly bounded by $a$, illustrating period collapse even for non-integral, and, in barycentric normalization, for irrational triangles (Fares, 20 Jan 2026).

6. Birational Geometry, Singularities, and Classification

Markov numbers classify certain degenerate surfaces in algebraic geometry. Hacking–Prokhorov proved that partial $-1$–Gorenstein smoothings of weighted projective planes $\mathbb{P}(a^2, b^2, c^2)$ arise if and only if $(a, b, c)$ is a Markov triangle (Urzúa et al., 2023). The Minimal Model Program (MMP) applied here proceeds by flips (Wahl singularities), organized into “Mori trains” tracked by continued-fraction expansions and explicit invariants:

• Numerical invariants: For each Markov number $c$, two weights $r_c$ and $w_c$ are defined ($r_c^2 \equiv -1 \pmod c$, $w_c \equiv 3a^{-1}b \pmod c$). These label combinatorial data for the classification.
• Stabilization: On any fixed branch, after three flips, the chain stabilizes at a unique antiflip. The total number of flips to reach the Fibonacci spine from any Markov triple is bounded above by $6\nu+3$ (with $\nu$ branch changes).
• Markov Conjecture: The largest entry $c$ in a Markov triple uniquely determines the other entries, equivalently, for each $c$ there are exactly two solutions to $r^2 \equiv -1 \pmod c$. Birational invariants encode this uniqueness.

These results reveal deep connections between combinatorial arithmetic, algebraic surface degenerations, and continued-fraction theory (Urzúa et al., 2023).

7. Periodicities and Palindromic Patterns

Markov region-numbers and associated special squares exhibit palindromic repeat cycles in their last digits along each edge. These cycles are Pisano-type periods nested within the Lucas sequence periods $U_k(3R, 1)$. In practice, the last $d$ digits form periodic sequences whose blocks concatenate into palindromic cycles. For region $R=5$, the two-digit cycle for $\Lambda_R(k)$ over $k=1, \dots, 12$ is $\{3, 13, 44, 75, 94, \dots\}$, whose reverse exactly matches the right-edge cycle, assembling into a full palindromic sequence of length 24. This periodic structure is uniform for all regions and persists irrespective of arithmetic normalization (Gore, 4 Jun 2025).