HEXAGONbeta

PARAMETER MONOTONICITY FOR GENERALIZED k-MARKOV NUMBERS

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.CO — Combinatorics

Contributed by Qiyue Tang ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

We prove that the order reversal asked about in Question 35 of Banaian and Huang cannot occur for generalized k-Markov numbers, answering their question negatively.

Provenance statement

The initial research draft was generated with ChatGPT-6-Pro from references and earlier research materials supplied by the author. The archived log records a complete proof draft within approximately 40 minutes of the session’s start. Codex assisted with mathematical review, exact symbolic checks, literature searches, manuscript organization, and English-language editing and polishing. The manuscript passed review by Rethlas-OSV .

References

  1. M. Aigner, Markov's Theorem and 100 Years of the Uniqueness Conjecture , Springer, Cham, 2013. doi:10.1007/978-3-319-00888-2 (https://doi.org/10.1007/978-3-319-00888-2).DOI
  2. M. Rabideau and R. Schiffler, Continued fractions and orderings on the Markov numbers , Adv. Math. 370 (2020), 107231. doi:10.1016/j.aim.2020.107231 (https://doi.org/10.1016/j.aim.2020.107231).DOI
  3. K. Lee, L. Li, M. Rabideau, and R. Schiffler, On the ordering of the Markov numbers , Adv. Appl. Math. 143 (2023), 102453. arXiv:2010.13010 (https://arxiv.org/abs/2010.13010).arXiv
  4. C. Lagisquet, E. Pelantová, S. Tavenas, and L. Vuillon, On the Markov numbers: fixed numerator, denominator, and sum conjectures , Adv. Appl. Math. 130 (2021), 102227. doi:10.1016/j.aam.2021.102227 (https://doi.org/10.1016/j.aam.2021.102227).DOI
  5. J. Gaster, Boundary slopes for the Markov ordering on relatively prime pairs , Adv. Math. 403 (2022), 108377. arXiv:2107.13499 (https://arxiv.org/abs/2107.13499).arXiv
  6. M. Huang, On the monotonicity of the generalized Markov numbers , arXiv:2204.11443v2 (https://arxiv.org/abs/2204.11443v2) (2022).arXiv
  7. E. Banaian and A. Sen, A generalization of Markov numbers , Ramanujan J. 63 (2024), 1021--1055. arXiv:2210.07366 (https://arxiv.org/abs/2210.07366).arXiv
  8. E. Banaian, Orderings on kk-Markov numbers , Ramanujan J. 71 (2026), article 12. doi:10.1007/s11139-026-01468-5 (https://doi.org/10.1007/s11139-026-01468-5).DOI
  9. E. Banaian and M. Huang, Orderings of Generalized kk-Markov Numbers , arXiv:2604.17445v1 (https://arxiv.org/abs/2604.17445v1) (2026).arXiv
  10. N. M. Doan, X. Li, and V. Nguyen, Stable-norm boundary geometry on once-punctured tori , arXiv:2605.14574v3 (https://arxiv.org/abs/2605.14574v3) (2026).arXiv
Show all 14 referencesShow fewer references
  1. Y. Gyoda and K. Matsushita, Generalization of Markov Diophantine equation via generalized cluster algebra , Electron. J. Combin. 30 (2023), no. 4, Paper 4.10, 20 pp.
  2. OSVerify contributors, Rethlas-OSV , software repository, 2026. https://github.com/OSVerify/Rethlas-OSV.link
  3. H. Ju, G. Gao, J. Jiang, B. Wu, Z. Sun, S. Liu, L. Chen, Yutong Wang, Yuefeng Wang, Z. Wang, W. He, P. Wu, L. Xiao, R. Liu, B. Dai, and B. Dong, Automated Conjecture Resolution with Formal Verification , arXiv:2604.03789v2 (https://arxiv.org/abs/2604.03789v2) (2026).arXiv
  4. F. Nie, H. Ye, S. Liu, Z. Wang, S. Ermon, and J. Zou, One-Sided Verification Under Limited Expert Review , arXiv:2610.18089v2 (https://arxiv.org/abs/2610.18089v2) (2026).arXiv

Version history

  1. v1Submitted by Qiyue TangInitial depositCurrentNov 02, 2026