Orderings of k-Markov Numbers

2025/12/03 by Esther Banaian, Banaian, Esther
Mathematics · #11B83 #13F60 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2512.04026

openalex publication_date 2025/12/03 · openalex created_date 2025/12/05 · openalex updated_date 2026/07/28

Abstract

The k-Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to x² + y² + z² + k(xy + xz + yz) = (3+3k)xyz. When k =0, this recovers the ordinary Markov numbers. A long-standing question in the theory of Markov numbers is Frobenius's unicity conjecture, concerning whether every Markov number is the maximum in a unique solution triple. Aigner gave a series of weaker, related conjectures which were confirmed to be true by Lee, Li, Rabideau, and Schiffler using techniques from the theory of cluster algebras. We show here that k-Markov numbers also satisfy Aigner's conjectures.

Citations

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Related

• Continued fractions and orderings on the Markov numbers
• On the Markov numbers: fixed numerator, denominator, and sum conjectures
• On the ordering of the Markov numbers
• Convexity and Aigner's Conjectures
• Uniqueness theorem of generalized Markov numbers that are prime powers