Uniqueness conjectures for extended Markov numbers

Abstract

We study an extension to the uniqueness conjecture for Markov numbers. For any three positive integers $m\geq a$ and $m\geq b$ satisfying $a^2+b^2+m^2=3abm$, this conjecture states that the triple $(a,m,b)$ is uniquely determined by the Markov number $m$. The theory of Markov numbers may be described by combinatorics of the sequences $(1,1)$ and $(2,2)$. There is an extension to the theory based on arbitrary sequences. We define extended uniqueness conjectures for any sequences $\mu$ and $\nu$. We show that for certain integers $a>1$ and $b>2$ the extended uniqueness conjecture for the sequences $\mu=(a,a)$ and $\nu=(b,b)$ fails.