The paper constructs generalized discrete Markov spectra for the family of equations x² + y² + z² + k1 yz + k2 zx + k3 xy = (3 + k1 + k2 + k3) xyz, with each spectrum element realized as both a Lagrange constant of a quadratic irrational and a Markov constant of an indefinite binary quadratic form.
Orderings of k-Markov Numbers
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
The $k$-Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. 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.
fields
math.NT 2verdicts
UNVERDICTED 2representative citing papers
Generalized k-Markov numbers grow monotonically along more random lines as k increases, supporting a k-analog of Frobenius' uniqueness conjecture.
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Generalized discrete Markov spectra
The paper constructs generalized discrete Markov spectra for the family of equations x² + y² + z² + k1 yz + k2 zx + k3 xy = (3 + k1 + k2 + k3) xyz, with each spectrum element realized as both a Lagrange constant of a quadratic irrational and a Markov constant of an indefinite binary quadratic form.
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Orderings of Generalized k-Markov Numbers
Generalized k-Markov numbers grow monotonically along more random lines as k increases, supporting a k-analog of Frobenius' uniqueness conjecture.