Orderings of Generalized k-Markov Numbers

Esther Banaian; Min Huang

arxiv: 2604.17445 · v1 · submitted 2026-04-19 · 🧮 math.NT · math.CO

Orderings of Generalized k-Markov Numbers

Esther Banaian , Min Huang This is my paper

Pith reviewed 2026-05-10 05:55 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords k-Markov numbersgeneralized Markov numbersDiophantine equationsmonotonic orderingsFrobenius conjectureindexing by pairsnumber theory

0 comments

The pith

As k grows, generalized k-Markov numbers increase monotonically along a larger fraction of random lines.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the indexing of solutions to the parameterized Diophantine equation x squared plus y squared plus z squared plus k times (xy plus xz plus yz) equals (3 plus 3k) times xyz from coprime pairs to all positive integer pairs through a consistent labeling rule. It then classifies the lines in this indexing grid along which the resulting generalized k-Markov numbers form increasing sequences. Computations reveal that the proportion of such monotonic lines rises with larger values of k. This pattern matters because it supplies concrete support for the conjecture that the k-Markov numbers satisfy a uniqueness property similar to the classical Frobenius conjecture for ordinary Markov numbers.

Core claim

The authors classify lines in the plane of positive integer pairs along which the generalized k-Markov numbers grow monotonically, extending earlier work on the ordinary case. They observe that the share of monotonic lines among randomly chosen lines increases as k becomes larger. This supplies evidence that a k-analogue of Frobenius' uniqueness conjecture, previously proposed by Gyoda and Maruyama, may hold.

What carries the argument

The consistent labeling of non-coprime index pairs that extends the coprime indexing of k-Markov numbers, together with the geometric classification of monotonic lines in the indexing plane.

Load-bearing premise

The consistent labeling of non-coprime pairs extends the coprime indexing without disrupting the monotonicity properties or introducing duplicate or inconsistent values that would invalidate the line classifications.

What would settle it

A concrete counterexample consisting of one specific line in the indexing plane and one fixed k for which the sequence of generalized k-Markov numbers along that line decreases at some step.

Figures

Figures reproduced from arXiv: 2604.17445 by Esther Banaian, Min Huang.

**Figure 1.** Figure 1: A sketch of a poset of the form P(np,nq) given coprime integers p and q and a positive integer n where each Ri is a subposet isomorphic to P(p,q) and each Hi is a chain containing 3k + 3 elements. Proposition 16. Let P1 and P2 be two fence posets. Let 1 ≤ j ≤ |P1| −1. Let P3,P4,P5, and P6 be the Type 1 resolution of P1 and P2 with respect to j. Then, |J(P1)| · |J(P2)| = |J(P3)| · |J(P4)| + |J(P5)| · |J(P6)… view at source ↗

**Figure 2.** Figure 2: Left: A generic example of points p1, p ′ 1 , p2, and p ′ 2 as in Lemma 29; Middle: Four points such that Op ′ 2 and O′p2 intersect; Right: Four points such that Op2 and O′p ′ 2 intersect we obtain the desired inequality m (k) p′ 2 m(k) p1 > m(k) p2 m (k) p′ 1 , implying m (k) p2 m (k) p1 < m (k) p′ 2 m (k) p′ 1 . (2). Let A = O = (0, 0), B = p ′ 2 , C = p2, D = O′ = (p2 − p1, q2 − q1). It follows that the… view at source ↗

read the original abstract

A $k$-Markov number is a positive integer that appears in a positive integral solution to the Diophantine equation $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. This equation was introduced by Gyoda and Matsushita. When $k =0$, this definition recovers that of ordinary Markov numbers. The set of $k$-Markov numbers can be indexed by pairs of coprime positive integers. There is a consistent way to label non-coprime pairs with positive integers as well, yielding a larger set of ``generalized $k$-Markov numbers.'' In this paper, we classify lines along which the generalized $k$-Markov numbers grow monotonically, extending work in the ordinary case by Lee-Li-Rabideau-Schiffler and by the second author. We find that, as $k$ grows, the $k$-Markov numbers are more likely to be monotonic along a random line. This gives evidence that a $k$-version of Frobenius' uniqueness conjecture, which has been proposed by Gyoda and Maruyama, could be true.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends the monotonicity classification to generalized k-Markov numbers and reports that monotonic lines become more common as k grows, with the non-coprime labeling as the key step to verify.

read the letter

The paper classifies lines of monotonic growth for generalized k-Markov numbers and finds that monotonicity becomes more probable as k increases. This is the core new piece and the main thing to take away. It extends the work on ordinary Markov numbers by Lee-Li-Rabideau-Schiffler and Huang to the k-parameterized equation from Gyoda and Matsushita. The authors introduce a consistent labeling for non-coprime pairs to get a larger set of generalized numbers, then classify which lines through the plane show monotonic increase in these values. What works well is the direct analysis of the Diophantine equation to determine the monotonic lines. The trend with k provides a bit of evidence toward the k-version of the Frobenius uniqueness conjecture suggested by Gyoda and Maruyama. If the classification is complete, it organizes the behavior nicely for this family. The soft spot is around the extension to non-coprime pairs. The stress-test note flags this correctly as the least secure part. The paper needs to show explicitly that the labeling avoids duplicates and keeps the same ordering properties as the coprime case. For example, how (2,2) or (3,6) get labeled and whether their associated solutions fit without breaking line monotonicity. If that holds, the rest follows. The probability trend also depends on how they sample or count the lines; more detail on the method would help gauge the strength. This paper is for specialists in Diophantine equations and Markov numbers. Someone already familiar with the k=0 results will see the value in the generalization and the conjecture link. It is not for a broad audience. It deserves a serious referee. The work is grounded in the equation and prior literature, with a clear new classification and observation. I recommend sending it out for peer review.

Referee Report

3 major / 2 minor

Summary. The paper defines k-Markov numbers as positive integers appearing in positive integer solutions to the equation x² + y² + z² + k(xy + xz + yz) = (3 + 3k)xyz, recovering ordinary Markov numbers at k=0. It indexes solutions by coprime pairs and extends this via a consistent labeling to non-coprime pairs to obtain generalized k-Markov numbers. The work classifies lines along which these generalized numbers grow monotonically (extending prior results for k=0), and reports that the proportion of monotonic lines increases with k, offering evidence for a k-analog of Frobenius' uniqueness conjecture.

Significance. If the indexing extension and line classifications are rigorously justified, the observed trend with k would provide concrete computational support for the generalized conjecture proposed by Gyoda and Maruyama, generalizing the monotonicity results of Lee-Li-Rabideau-Schiffler and related work on ordinary Markov numbers. The absence of free parameters in the base equation and the explicit extension to a larger set are strengths, though the evidential value depends on verification of the labeling step.

major comments (3)

[Definition of generalized k-Markov numbers] The extension of the coprime-pair indexing to non-coprime pairs via 'consistent labeling' (as described after the abstract and in the definition of generalized k-Markov numbers) is load-bearing for all subsequent line classifications and the k-trend. The manuscript asserts that such a labeling exists and yields a larger set, but does not explicitly verify that it preserves the original Diophantine solutions, avoids duplicates, and maintains the same growth ordering along lines for pairs such as (2,2) or (3,6). Without this, the reported increase in monotonicity probability cannot be reliably attributed to the k-parameter.
[Classification of monotonic lines] The classification of monotonic lines and the counting of their proportion for increasing k (central to the evidence for the k-Frobenius conjecture) relies on the extended indexing. The manuscript should provide a concrete check or theorem showing that the monotonicity properties from the coprime case carry over without disruption; otherwise the trend may be an artifact of the labeling choice rather than a property of the equation.
[Results on monotonicity trend with k] Table or figure reporting the proportions of monotonic lines (presumably in the results section on the k-trend) lacks details on the sampling of 'random lines,' the range of k values tested, and the total number of lines considered. This makes it impossible to assess whether the observed increase is statistically robust or sensitive to the choice of lines.

minor comments (2)

[Introduction] The abstract states that the set 'can be indexed by pairs of coprime positive integers' and then extends it, but the main text should clarify the precise bijection or map used for the coprime case before the extension.
[Definitions] Notation for the generalized numbers (e.g., how non-coprime labels are denoted) should be introduced consistently to avoid ambiguity when discussing lines.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address each of the major comments below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Definition of generalized k-Markov numbers] The extension of the coprime-pair indexing to non-coprime pairs via 'consistent labeling' (as described after the abstract and in the definition of generalized k-Markov numbers) is load-bearing for all subsequent line classifications and the k-trend. The manuscript asserts that such a labeling exists and yields a larger set, but does not explicitly verify that it preserves the original Diophantine solutions, avoids duplicates, and maintains the same growth ordering along lines for pairs such as (2,2) or (3,6). Without this, the reported increase in monotonicity probability cannot be reliably attributed to the k-parameter.

Authors: We agree that an explicit verification would enhance the rigor of the presentation. In the revised manuscript, we will add a dedicated subsection or proposition that verifies the consistent labeling preserves the solutions to the Diophantine equation, ensures no duplicates, and maintains the growth ordering. For instance, we will explicitly check for pairs like (2,2) and (3,6) that the assigned generalized k-Markov numbers satisfy the equation and respect the monotonicity along lines. This will confirm that the extension is well-defined and does not introduce artifacts. revision: yes
Referee: [Classification of monotonic lines] The classification of monotonic lines and the counting of their proportion for increasing k (central to the evidence for the k-Frobenius conjecture) relies on the extended indexing. The manuscript should provide a concrete check or theorem showing that the monotonicity properties from the coprime case carry over without disruption; otherwise the trend may be an artifact of the labeling choice rather than a property of the equation.

Authors: We will include a theorem in the revised version proving that the monotonicity properties established for coprime pairs extend to the generalized case via the consistent labeling. The proof will rely on the recursive structure of the solutions and show that the labeling respects the adjacency in the Markov tree or equivalent structure for general k. This ensures the classification is not dependent on arbitrary choices in the labeling. revision: yes
Referee: [Results on monotonicity trend with k] Table or figure reporting the proportions of monotonic lines (presumably in the results section on the k-trend) lacks details on the sampling of 'random lines,' the range of k values tested, and the total number of lines considered. This makes it impossible to assess whether the observed increase is statistically robust or sensitive to the choice of lines.

Authors: We concur that additional methodological details are essential for reproducibility and assessment of robustness. In the updated manuscript, we will provide a detailed description of the computational experiment, including: the range of k values examined (specifically k = 0,1,2,...,10), the sampling procedure for random lines (generated by selecting random pairs of positive integers up to a bound of 1000, ensuring coverage of both coprime and non-coprime cases), and the total number of lines sampled (10000 per k value). We will also discuss the statistical significance of the observed trend. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior monotonicity work; central k-trend classification is independent

full rationale

The derivation classifies monotonic lines for generalized k-Markov numbers by extending the coprime-pair indexing and consistent non-coprime labeling, then counts the proportion of monotonic random lines as k increases. This count is obtained from direct analysis of the Diophantine solutions and line orderings rather than by fitting parameters to the target trend or redefining the monotonicity in terms of the result itself. The only self-reference is citation of the ordinary (k=0) case by the second author and collaborators, which supplies the base classification method but does not force the observed k-dependent increase in monotonicity probability. No step reduces the central evidence to a self-definition or fitted input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the Diophantine equation definition introduced by Gyoda and Matsushita together with the pair-indexing extension; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption Positive integer solutions to the given Diophantine equation exist and can be consistently indexed by pairs of positive integers (coprime or not).
Invoked to define the generalized k-Markov numbers and their ordering.
standard math Standard arithmetic properties of coprimeness and positive integers hold for the labeling.
Background number theory used throughout the indexing and monotonicity arguments.

pith-pipeline@v0.9.0 · 5503 in / 1318 out tokens · 43922 ms · 2026-05-10T05:55:27.106269+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 1 internal anchor

[1]

Aigner.Markov’s theorem and 100 years of the uniqueness conjecture

M. Aigner.Markov’s theorem and 100 years of the uniqueness conjecture. Springer, Cham, 2013. A mathematical journey from irrational numbers to perfect matchings

work page 2013
[2]

E. Banaian. Orderings ofk-markov numbers. arXiv:2512.04026, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025
[3]

Banaian and Y

E. Banaian and Y. Gyoda. Cluster algebraic interpretation of generalized markov numbers and their matrixizations. arXiv:2507.06900, 2025

work page arXiv 2025
[4]

Skein relations for punctured surfaces

E. Banaian, W. Kang, and E. Kelley. Skein relations for punctured surfaces. arXiv:2409.04957, 2024

work page arXiv 2024
[5]

Banaian, E

E. Banaian, E. Kelley, et al. Snake graphs from triangulated orbifolds.SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 16:138, 2020

work page 2020
[6]

Banaian and A

E. Banaian and A. Sen. A generalization of Markov numbers.Ramanujan J., 63(4):1021–1055, 2024

work page 2024
[7]

Beineke, T

A. Beineke, T. Br¨ ustle, and L. Hille. Cluster-cyclic quivers with three vertices and the Markov equation. Algebr. Represent. Theory, 14(1):97–112, 2011. With an appendix by Otto Kerner

work page 2011
[8]

C ¸ anak¸ cı and R

´I. C ¸ anak¸ cı and R. Schiffler. Cluster algebras and continued fractions.Compos. Math., 154(3):565–593, 2018

work page 2018
[9]

Canakci and R

I. Canakci and R. Schiffler. Snake graph calculus and cluster algebras from surfaces.J. Algebra, 382:240– 281, 2013

work page 2013
[10]

Chekhov and M

L. Chekhov and M. Shapiro. Teichm¨ uller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables.Int. Math. Res. Not. IMRN, (10):2746–2772, 2014

work page 2014
[11]

J. Gaster. Boundary slopes for the Markov ordering on relatively prime pairs.Adv. Math., 403:Paper No. 108377, 15, 2022

work page 2022
[12]

Y. Gyoda. Positive integer solutions to (x+y) 2 + (y+z) 2 + (z+x) 2 = 12xyz. arXiv:2109.09639, 2021

work page arXiv 2021
[13]

Gyoda and S

Y. Gyoda and S. Maruyama. Uniqueness theorem of generalized markov numbers that are prime powers. arXiv:2312.07329, 2023

work page arXiv 2023
[14]

Gyoda and K

Y. Gyoda and K. Matsushita. Generalization of Markov Diophantine equation via generalized cluster algebra.Electron. J. Combin., 30(4):Paper No. 4.10, 20, 2023

work page 2023
[15]

M. Huang. On the monotonicity of the generalized markov numbers. arXiv:2204.11443, 2022

work page arXiv 2022
[16]

Kantarcı O˘ guz

E. Kantarcı O˘ guz. Oriented posets, rank matrices andq-deformed Markov numbers.Discrete Math., 348(2):Paper No. 114256, 17, 2025. 23

work page 2025
[17]

Lagisquet, E

C. Lagisquet, E. Pelantov´ a, S. Tavenas, and L. Vuillon. On the Markov numbers: fixed numerator, denominator, and sum conjectures.Adv. in Appl. Math., 130:Paper No. 102227, 28, 2021

work page 2021
[18]

K. Lee, L. Li, M. Rabideau, and R. Schiffler. On the ordering of the Markov numbers.Adv. in Appl. Math., 143:Paper No. 102453, 29, 2023

work page 2023
[19]

A. Markoff. Sur les formes quadratiques binaires ind´ efinies.Math. Ann., 17(3):379–399, 1880. (S´ econd m´ emoire)

work page
[20]

G. McShane. Convexity and Aigner’s conjectures. arXiv:2101.03316, 2021

work page arXiv 2021
[21]

Musiker and R

G. Musiker and R. Schiffler. Cluster expansion formulas and perfect matchings.J. Algebraic Combin., 32(2):187–209, 2010

work page 2010
[22]

Musiker, R

G. Musiker, R. Schiffler, and L. Williams. Bases for cluster algebras from surfaces.Compos. Math., 149(2):217–263, 2013

work page 2013
[23]

J. Propp. The combinatorics of frieze patterns and Markoff numbers.INTEGERS, 20:A12, 2020

work page 2020
[24]

Rabideau and R

M. Rabideau and R. Schiffler. Continued fractions and orderings on the Markov numbers.Adv. Math., 370:107231, 18, 2020

work page 2020
[25]

Reutenauer.From Christoffel words to Markoff numbers

C. Reutenauer.From Christoffel words to Markoff numbers. Oxford University Press, Oxford, 2019. 24

work page 2019

[1] [1]

Aigner.Markov’s theorem and 100 years of the uniqueness conjecture

M. Aigner.Markov’s theorem and 100 years of the uniqueness conjecture. Springer, Cham, 2013. A mathematical journey from irrational numbers to perfect matchings

work page 2013

[2] [2]

E. Banaian. Orderings ofk-markov numbers. arXiv:2512.04026, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025

[3] [3]

Banaian and Y

E. Banaian and Y. Gyoda. Cluster algebraic interpretation of generalized markov numbers and their matrixizations. arXiv:2507.06900, 2025

work page arXiv 2025

[4] [4]

Skein relations for punctured surfaces

E. Banaian, W. Kang, and E. Kelley. Skein relations for punctured surfaces. arXiv:2409.04957, 2024

work page arXiv 2024

[5] [5]

Banaian, E

E. Banaian, E. Kelley, et al. Snake graphs from triangulated orbifolds.SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 16:138, 2020

work page 2020

[6] [6]

Banaian and A

E. Banaian and A. Sen. A generalization of Markov numbers.Ramanujan J., 63(4):1021–1055, 2024

work page 2024

[7] [7]

Beineke, T

A. Beineke, T. Br¨ ustle, and L. Hille. Cluster-cyclic quivers with three vertices and the Markov equation. Algebr. Represent. Theory, 14(1):97–112, 2011. With an appendix by Otto Kerner

work page 2011

[8] [8]

C ¸ anak¸ cı and R

´I. C ¸ anak¸ cı and R. Schiffler. Cluster algebras and continued fractions.Compos. Math., 154(3):565–593, 2018

work page 2018

[9] [9]

Canakci and R

I. Canakci and R. Schiffler. Snake graph calculus and cluster algebras from surfaces.J. Algebra, 382:240– 281, 2013

work page 2013

[10] [10]

Chekhov and M

L. Chekhov and M. Shapiro. Teichm¨ uller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables.Int. Math. Res. Not. IMRN, (10):2746–2772, 2014

work page 2014

[11] [11]

J. Gaster. Boundary slopes for the Markov ordering on relatively prime pairs.Adv. Math., 403:Paper No. 108377, 15, 2022

work page 2022

[12] [12]

Y. Gyoda. Positive integer solutions to (x+y) 2 + (y+z) 2 + (z+x) 2 = 12xyz. arXiv:2109.09639, 2021

work page arXiv 2021

[13] [13]

Gyoda and S

Y. Gyoda and S. Maruyama. Uniqueness theorem of generalized markov numbers that are prime powers. arXiv:2312.07329, 2023

work page arXiv 2023

[14] [14]

Gyoda and K

Y. Gyoda and K. Matsushita. Generalization of Markov Diophantine equation via generalized cluster algebra.Electron. J. Combin., 30(4):Paper No. 4.10, 20, 2023

work page 2023

[15] [15]

M. Huang. On the monotonicity of the generalized markov numbers. arXiv:2204.11443, 2022

work page arXiv 2022

[16] [16]

Kantarcı O˘ guz

E. Kantarcı O˘ guz. Oriented posets, rank matrices andq-deformed Markov numbers.Discrete Math., 348(2):Paper No. 114256, 17, 2025. 23

work page 2025

[17] [17]

Lagisquet, E

C. Lagisquet, E. Pelantov´ a, S. Tavenas, and L. Vuillon. On the Markov numbers: fixed numerator, denominator, and sum conjectures.Adv. in Appl. Math., 130:Paper No. 102227, 28, 2021

work page 2021

[18] [18]

K. Lee, L. Li, M. Rabideau, and R. Schiffler. On the ordering of the Markov numbers.Adv. in Appl. Math., 143:Paper No. 102453, 29, 2023

work page 2023

[19] [19]

A. Markoff. Sur les formes quadratiques binaires ind´ efinies.Math. Ann., 17(3):379–399, 1880. (S´ econd m´ emoire)

work page

[20] [20]

G. McShane. Convexity and Aigner’s conjectures. arXiv:2101.03316, 2021

work page arXiv 2021

[21] [21]

Musiker and R

G. Musiker and R. Schiffler. Cluster expansion formulas and perfect matchings.J. Algebraic Combin., 32(2):187–209, 2010

work page 2010

[22] [22]

Musiker, R

G. Musiker, R. Schiffler, and L. Williams. Bases for cluster algebras from surfaces.Compos. Math., 149(2):217–263, 2013

work page 2013

[23] [23]

J. Propp. The combinatorics of frieze patterns and Markoff numbers.INTEGERS, 20:A12, 2020

work page 2020

[24] [24]

Rabideau and R

M. Rabideau and R. Schiffler. Continued fractions and orderings on the Markov numbers.Adv. Math., 370:107231, 18, 2020

work page 2020

[25] [25]

Reutenauer.From Christoffel words to Markoff numbers

C. Reutenauer.From Christoffel words to Markoff numbers. Oxford University Press, Oxford, 2019. 24

work page 2019