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.
McShane,Convexity and Aigner’s Conjectures, 2021
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Generalized k-Markov numbers grow monotonically along more random lines as k increases, supporting a k-analog of Frobenius' uniqueness conjecture.
k-Markov numbers satisfy Aigner's conjectures on their orderings and uniqueness properties in positive integer solutions.
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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.
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Orderings of k-Markov Numbers
k-Markov numbers satisfy Aigner's conjectures on their orderings and uniqueness properties in positive integer solutions.