Uniqueness conjectures for extended Markov numbers

TL;DR

This paper investigates an extension of the Markov number uniqueness conjecture, demonstrating that for certain sequences, the extended conjecture does not hold, thus revealing limitations of the generalized theory.

Contribution

It introduces extended uniqueness conjectures for Markov numbers based on arbitrary sequences and shows counterexamples where these conjectures fail.

Findings

Extended conjectures do not always hold for specific sequences.

Counterexamples are provided for sequences ,a,b.

The classical uniqueness conjecture is limited in its generalization.

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.