Markoff Equation and Nilpotent Matrices
arXiv:0709.1499
Abstract
A triple (a,b,c) of positive integers is called a Markoff triple iff it satisfies the diophantine equation a2 + b2 + c2 = abc . Recasting the Markoff tree, whose vertices are Markoff triples, in the framework of intergral upper triangular 3x3 matrices, it will be shown that the largest member of such a triple determines the other two uniquely. This answers a question which has been open for almost 100 years.
Most of the (correct) portion of this paper has been incorporated into the paper "On the Markoff equation" (arXiv:1208.4032)