Markov's equation is not partition regular

2025/10/11 by Tao, Tianyi, Yang, Bohan
#05D10 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.10151

Abstract

Markov's equation x² + y² + z² = 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.

Citations

• Partition regularity of Pythagorean pairs
• Ramsey properties of nonlinear Diophantine equations
• Fermat-like equations that are not partition regular
• Higher order Fourier analysis of multiplicative functions and applications

Related

• Uniqueness theorem of generalized Markov numbers that are prime powers
• Continued fractions and orderings on the Markov numbers
• On the ordering of the Markov numbers
• On the Markov numbers: fixed numerator, denominator, and sum conjectures
• Orderings of k-Markov Numbers