Orderings of Generalized k-Markov Numbers

Abstract: A $k$-Markov number is a positive integer that appears in a positive integral solution to the Diophantine equation $$x² + y² + z² + k(xy + xz + yz) = (3+3k)xyz$$. 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.

• The paper introduces a classification of monotonicity in generalized k-Markov numbers along lattice lines using precise thresholds L(k) and U(k).
• It employs combinatorial fence posets and snake graphs to relate number properties with continued fractions and recurrence relations.
• Results indicate that as k increases, non-monotonic regions vanish, supporting a k-analogue of the classical Markov Uniqueness Conjecture.

Orderings of Generalized $k$-Markov Numbers

Introduction and Context

The paper "Orderings of Generalized $k$-Markov Numbers" (2604.17445) systematically advances the study of the arithmetic and combinatorial structure of Markov-type numbers arising from a class of Diophantine equations. While the classical Markov numbers are defined by solutions to the equation $x^2 + y^2 + z^2 = 3xyz$, the generalized $k$-Markov numbers are obtained from solutions to $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$ for a fixed nonnegative integer $k$—a deformation introduced by Gyoda and Matsushita.

A major focus in the literature has been the Uniqueness Conjecture (Frobenius, 1913) for classical Markov numbers, which posits that each Markov number is the largest in a unique Markov triple. Recent work has verified several weaker conjectures related to orderings and monotonicity in the Markov spectrum, notably along lines in the parameter space, and begun extending this to both "generalized Markov numbers" (indices by non-coprime integer pairs) and further to $k$-Markov analogues. This paper makes substantial progress in these directions, providing a detailed classification of monotonicity and orderings of generalized $k$-Markov numbers along lines in $\mathbb{Z}^2$.

Generalized $k$-Markov Numbers: Definitions and Combinatorial Framework

The $k$0-Markov numbers are positive integers occurring as entries in positive integer solutions of the $k$1-Markov equation. The construction extends to a larger family indexed by all pairs of positive integers via a correspondence with snake graphs—combinatorial objects originally developed in the context of cluster algebras from surfaces.

Each pair $k$2 gives rise to a "fence poset" $k$3, an oriented Hasse diagram constructed by tracing appropriately perturbed lattice segments according to specified combinatorial rules designed to avoid pathological collinearities. The generalized $k$4-Markov number $k$5 is defined as the number of order ideals of this fence poset, which, based on its combinatorial structure, reduces to computing the numerator of a specific continued fraction determined by the "shape" of the poset.

For coprime $k$6, these numbers recover the $k$7-Markov number $k$8, with $k$9 corresponding to the classical Markov spectrum.

Monotonicity and Orderings along Lines

A principal result of the study is a full classification of lines in $x^2 + y^2 + z^2 = 3xyz$0 along which the generalized $x^2 + y^2 + z^2 = 3xyz$1-Markov numbers are monotonic in $x^2 + y^2 + z^2 = 3xyz$2. The lines are of the form $x^2 + y^2 + z^2 = 3xyz$3 with rational $x^2 + y^2 + z^2 = 3xyz$4. Detailed analysis using recurrence relations, continued fraction expansions, and poset-theoretic skein relations leads to sharp asymptotic thresholds for monotonicity, depending on $x^2 + y^2 + z^2 = 3xyz$5.

For the key theorem, precise critical slopes $x^2 + y^2 + z^2 = 3xyz$6 and $x^2 + y^2 + z^2 = 3xyz$7 are given:

• If $x^2 + y^2 + z^2 = 3xyz$8, the sequence of $x^2 + y^2 + z^2 = 3xyz$9-Markov numbers along $k$0 is strictly increasing.
• If $k$1, the sequence is strictly decreasing.
• For $k$2, the sequence is not monotone; rather, it decreases and then increases.

These thresholds are given explicitly via expressions involving $k$3 and certain recursively defined sequences (generalizations of Fibonacci and Pell numbers) that asymptotically control the ordering phenomenon. Numerical data show that as $k$4 increases, the difference $k$5 shrinks toward zero, so the interval of non-monotonic ("gray zone") slopes vanishes in the large-$k$6 limit.

Main Theorem (Summarized)

Let $k$7 (coprime positive $k$8) and let $k$9 be defined as in the paper. For any line $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$0:

• If $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$1: $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$2 increases as $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$3 increases along $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$4.
• If $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$5: $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$6 decreases as $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$7 increases.
• Otherwise, the sequence transitions from decreasing to increasing.

The proof utilizes both recurrence relations for special subsequences (generalized Fibonacci/Pell) and a sophisticated analysis of ratios of generalized $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$8-Markov numbers at adjacent lattice points, combined with (generalized) Ptolemy-type inequalities for the associated combinatorial posets.

Theoretical and Structural Implications

This classification subsumes previous proofs for weaker conjectures for $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$9 (Aigner's constant sum/numerator/denominator conjectures proved by several teams: e.g., [LLRS], [LPTV], [mcshane2021convexity]), explicates the geometric-combinatorial nature of the fence poset correspondences in the $k$0 setting, and rigorously extends the theory to include the entire non-coprime locus.

A central finding is that as $k$1, the region of non-monotonicity collapses, and monotonic behavior becomes generic, lending strong evidence to a $k$2-analogue of the Uniqueness Conjecture for Markov numbers proposed by Gyoda and Maruyama [gyoda2023uniqueness].

The paper also provides examples illustrating that the total order structures induced by $k$3-Markov numbers may differ for various $k$4 in the "gray zone," but for sufficiently large $k$5, orderings stabilize.

Practical Implications and Future Directions

The developments here provide explicit tools for determining the local and global ordering of generalized $k$6-Markov numbers, with applications across arithmetic geometry (Diophantine approximation spectra, Markoff surfaces), combinatorics (continued fractions, lattice posets), and cluster algebras (combinatorial categorification via snake graphs on triangulated surfaces).

Practically, this means:

• For any $k$7 and any line in $k$8, one can algorithmically determine the ordering of the $k$9-Markov numbers along that line.
• For large $k$0, generic orderings become predictable, and the classical uniqueness heuristic applies more broadly.
• The combinatorial machinery constructed (fence posets, extended snake graphs) enables further connections with cluster algebra theory and poset combinatorics.

Open Problems

The paper concludes with conjectures regarding injectivity of the map $k$1 for all $k$2, and the stabilization of orderings for large $k$3. An explicit question is posed about the existence (or not) of pairs that reverse orderings as $k$4 varies in the overlap of their non-monotonic intervals, with current evidence suggesting stabilization after a certain $k$5 threshold.

Conclusion

This paper delivers a definitive structural description of the orderings for generalized $k$6-Markov numbers, giving explicit monotonicity regions, asymptotic results, and practical tools for their computation and comparison. The results not only generalize earlier work on classical Markov numbers but also provide significant evidence toward the extension of the Uniqueness Conjecture in the $k$7-deformed case. The methods developed will be broadly relevant for future work in both number theory and combinatorial algebra, particularly in the study of generalizations of Markov spectra and their cluster algebraic interpretations.