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Markov Number Graphs Extended to all Integer Triples

Spencer Scutt, Mark Turpin

TL;DR

This work investigates graphs generated by Vieta jumping from arbitrary integer triples, extending the classical Markov-triple framework. It introduces seeds, norms, and bases to analyze the resulting graphs, proving that exactly nine equivalence classes arise under undirected graph isomorphism, each with distinct structural signatures including infinite binary-tree branches and specific base configurations. The classification is accomplished through careful base-case analysis and two key lemmas that establish norm growth and sign-symmetry, yielding a comprehensive taxonomy of the corresponding graphs. The results generalize Markov dynamics to a broader Diophantine setting, illuminate connections to generalized equations $x^2+y^2+z^2=3xyz+k$, and motivate future exploration into real, rational, and complex extensions as well as other quadratic Vieta relations.

Abstract

We study the graphs generated when the formula for linking Markov triples is applied to general triples of integers. We find there are a finite number of equivalence classes of graphs, each with particular properties.

Markov Number Graphs Extended to all Integer Triples

TL;DR

This work investigates graphs generated by Vieta jumping from arbitrary integer triples, extending the classical Markov-triple framework. It introduces seeds, norms, and bases to analyze the resulting graphs, proving that exactly nine equivalence classes arise under undirected graph isomorphism, each with distinct structural signatures including infinite binary-tree branches and specific base configurations. The classification is accomplished through careful base-case analysis and two key lemmas that establish norm growth and sign-symmetry, yielding a comprehensive taxonomy of the corresponding graphs. The results generalize Markov dynamics to a broader Diophantine setting, illuminate connections to generalized equations , and motivate future exploration into real, rational, and complex extensions as well as other quadratic Vieta relations.

Abstract

We study the graphs generated when the formula for linking Markov triples is applied to general triples of integers. We find there are a finite number of equivalence classes of graphs, each with particular properties.
Paper Structure (22 sections, 3 theorems, 31 equations, 1 table)

This paper contains 22 sections, 3 theorems, 31 equations, 1 table.

Table of Contents

  1. Introduction
  2. Arbitrary Application of the Method
  3. Fundamental Ideas
  4. Seeds
  5. Norm of a Vertex
  6. Bases
  7. Graph equivalence
  8. Infinite Binary Tree
  9. The Main Theorem
  10. Two important lemmas
  11. Classification
  12. Base $(0,0,0)$
  13. Base $(0,0,a)$
  14. Base $(0,a,b)$
  15. Base $(0,a,a)$ or $(0,-a,a)$
  16. ...and 7 more sections

Key Result

Theorem 1

There exist exactly nine equivalence classes of graphs formed by the Markov Vieta jumpings applied to unordered triples of integers.

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof