Markov uniqueness conjecture for Markov triples
Prove that every Markov triple of positive integers satisfying x^2+y^2+z^2=3xyz is uniquely determined by its largest entry, equivalently that no two distinct Markov triples have the same Markov number.
References
A triple of positive integers $x,y,z>0$ such that $x2+y2+z2=3xyz$ is called a Markov triple and were introduced by Markov in the late XIX century. Frobenius conjectured in that these triples are always determined by their largest number, also called a Markov number. This is known as Markov's uniqueness conjecture, see . Since then, many partial results have been made for Markov numbers of a given form, such as prime powers or particular linear functions of prime powers, see e.g. . However, the general statement remains open.
A triple of positive integers $x,y,z>0$ such that $x2+y2+z2=3xyz$ is called a Markov triple and were introduced by Markov in the late XIX century. Frobenius conjectured in that these triples are always determined by their largest number, also called a Markov number. This is known as Markov's uniqueness conjecture, see . Since then, many partial results have been made for Markov numbers of a given form, such as prime powers or particular linear functions of prime powers, see e.g. . However, the general statement remains open.
A triple of positive integers $x,y,z>0$ such that $x2+y2+z2=3xyz$ is called a Markov triple and were introduced by Markov in the late XIX century. Frobenius conjectured in that these triples are always determined by their largest number, also called a Markov number. This is known as Markov's uniqueness conjecture, see . Since then, many partial results have been made for Markov numbers of a given form, such as prime powers or particular linear functions of prime powers, see e.g. . However, the general statement remains open.