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Almost all Markoff numbers are unique.

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by William Y. Chen

Version 2 / Oct 16, 2026 / CC BY 4.0

Abstract

A Markoff number is the largest coordinate of a positive integer solution of a2+b2+c2=3abca^2+b^2+c^2=3abc. Frobenius conjectured in 1913 that every Markoff number is the largest coordinate of exactly one such solution with a≤b≤ca\le b\le c. We prove that the conjecture holds for almost all Markoff numbers: the proportion of exceptions among the Markoff numbers up to XX tends to 00 as X→∞X\to\infty. Two such solutions with the same largest coordinate give simple closed geodesics of equal length on the modular torus, that is, primitive homology classes v,wv,w with equal values N(v)=N(w)N(v)=N(w) of the McShane--Rivin stable norm NN. Estimates for the boundary of the unit ball of NN reduce the problem to the equations N(Gv)=N(v)N(Gv)=N(v) with GG a rational linear map. For each GG that is not a symmetry of the torus, a zero-one law for the 33-adic analytic dynamics of the mapping class group on the Markoff surface shows that the solutions vv have density zero.

Provenance statement

This project was primarily completed using ChatGPT Astra via Codex, and Claude Fable 5.1, using skills developed by the author and Taylor Dupuy. The author initiated the project and gave the model some of his ideas for approaching the problem, but the argument as it appears here is largely due to Astra.

Tools used

OpenAI
CodexVersion Astra
Anthropic
Claude FableVersion 5.1

References

No bibliography entries or stable reference identifiers were extracted from this source.

Version history

  1. v1Submitted by William Y. ChenInitial depositSupersededOct 16, 2026
  2. v2Submitted by William Y. ChenSubmitted as a new versionCurrentOct 16, 2026