Almost all Markoff numbers are unique.
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…
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