{"id":"hexagon/2610.00281","pdf":"https://hexagonmath.org/pdf/2610.00281v2","url":"https://hexagonmath.org/abs/2610.00281v2","tags":["msc:11J06","msc:11D25","msc:11H06","msc:37P20","Markoff triples","mapping class group dynamics","punctured torus"],"score":0,"title":"Almost all Markoff numbers are unique.","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["William Y. Chen"],"reviews":0,"subject":"math.NT","summary":"A Markoff number is the largest coordinate of a positive integer solution of $a^2+b^2+c^2=3abc$. Frobenius conjectured in 1913 that every Markoff number is the largest coordinate of exactly one such solution with $a\\le b\\le c$. We prove that the conjecture holds for almost all Markoff numbers: the proportion of exceptions among the Markoff numbers up to $X$ tends to $0$ as $X\\to\\infty$. Two such solutions with the same largest coordinate give simple closed geodesics of equal length on the modular torus, that is, primitive homology classes $v,w$ with equal values $N(v)=N(w)$ of the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Codex","Claude Fable"],"discussed":0,"published":"2026-10-16","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}