Markov's equation is not partition regular
Tianyi Tao, Bohan Yang
TL;DR
Problem: determine whether the Markov equation $x^2 + y^2 + z^2 = 3xyz$ is partition regular over $\mathbb{N}$. Approach: a constructive 9-coloring argument that leverages the equation's structure; the sets $A_i$, $B_i$, and $C$ are used to rule out homochromatic nontrivial solutions by modular considerations. Key contributions: a complete proof that the equation is not partition regular without assuming the Uniqueness Conjecture, with a framework that combines modular methods and the intrinsic Markov structure; this provides a necessary condition related to the Uniqueness Conjecture. Significance: establishes rigidity-like obstructions for partition regularity of Markov-type Diophantine equations and provides techniques potentially applicable to related inhomogeneous problems.
Abstract
Markov's equation x^2 + y^2 + z^2 = 3xyz is a widely studied topic in number theory, and the structure of its solutions has profound connections with mathematical fields such as combinatorics, hyperbolic geometry, approximation theory, and cluster algebras. In this paper, we prove that Markov's equation is not partition regular, which also confirms a necessary condition for the Uniqueness Conjecture.