Patterns in the Markov numbers and their generalizations
Abstract
Positive integer solutions to x2+y2+z2− x y z =D correspond to the important Markov (Markoff) numbers when D=0. From a given solution triple, three more are found with Vieta involutions, making an infinite tree of solutions. Starting instead with three real numbers greater than $2$ gives lengths of closed geodesics in a punctured torus. In this paper we study these tree structures for any real D. A continuous function, originally related to a norm on homology, encodes all the numbers on each of these trees. The properties of this function are developed here in general, with a self-contained exposition, showing that the usual D=0 case is part of a bigger picture. Encoding function graphs are shown to be convex for D<4, straight lines for D=4 and concave for D>4. Among other consequences are generalizations to all D of: estimates for counting numbers on these trees, descriptions of the geometry of the corresponding lattice curves, uniqueness conditions, and identities of McShane and Hines.
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BibTeX
@misc{osullivan2026patterns,
title = {{Patterns in the Markov numbers and their generalizations}},
author = {Cormac O'Sullivan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14149},
eprint = {2609.14149},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14149},
}
RIS
TY - UNPB
TI - Patterns in the Markov numbers and their generalizations
AU - Cormac O'Sullivan
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14149
UR - https://arxiv.org/abs/2609.14149
AB - Positive integer solutions to x^2+y^2+z^2− x y z =D correspond to the important Markov (Markoff) numbers when D=0. From a given solution triple, three more are found with Vieta involutions, making an infinite tree of solutions. Starting instead with three real numbers greater than $2$ gives lengths of closed geodesics in a punctured torus. In this paper we study these tree structures for any real D. A continuous function, originally related to a norm on homology, encodes all the numbers on each of these trees. The properties of this function are developed here in general, with a self-contained exposition, showing that the usual D=0 case is part of a bigger picture. Encoding function graphs are shown to be convex for D<4, straight lines for D=4 and concave for D>4. Among other consequences are generalizations to all D of: estimates for counting numbers on these trees, descriptions of the geometry of the corresponding lattice curves, uniqueness conditions, and identities of McShane and Hines.
ER -