On the Markov numbers: fixed numerator, denominator, and sum conjectures

TL;DR

This paper proves two conjectures related to the ordering and uniqueness of Markov numbers, generalizes them to non-prime pairs, and confirms the conjectures for this broader set, enhancing understanding of their structure.

Contribution

It proves two of Aigner's conjectures about Markov numbers, generalizes the concept to non-prime pairs, and confirms the conjectures for this extended set.

Findings

Proved two conjectures about Markov numbers' order and uniqueness.

Generalized Markov numbers to pairs (p,q) with p ≤ q.

Confirmed the conjectures hold for the generalized set.

Abstract

The Markov numbers are the positive integer solutions of the Diophantine equation $x^2 + y^2 + z^2 = 3xyz$. Already in 1880, Markov showed that all these solutions could be generated along a binary tree. So it became quite usual (and useful) to index the Markov numbers by the rationals between 0 and 1 which stand at the same place in the Stern-Brocot binary tree. The Frobenius conjecture claims that each Markov number appears at most once in the tree. In particular, if the conjecture is true, the order of Markov numbers would establish a new strict order on the rationals. Aigner suggested three conjectures to better understand this order. The first one has already been solved for a few months. We prove that the other two conjectures are also true. Along the way, we generalize Markov numbers to any couple (p,q) of nonnegative integers (not only when they are relatively primes) and conjecture that the unicity is still true as soon as $p \leq q$. Finally, we show that the three conjectures are in fact true for this superset.