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The Markov numbers are the positive integer solutions of the Diophantineequation $x^2 + y^2 + z^2 = 3xyz$. Already in 1880, Markov showed that allthese solutions could be generated along a binary tree. So it became quiteusual (and useful) to index the Markov numbers by the rationals between 0 and 1which stand at the same place in the Stern-Brocot binary tree. The Frobeniusconjecture claims that each Markov number appears at most once in the tree.  In particular, if the conjecture is true, the order of Markov numbers wouldestablish a new strict order on the rationals. Aigner suggested threeconjectures to better understand this order. The first one has already beensolved 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) ofnonnegative integers (not only when they are relatively primes) and conjecturethat the unicity is still true as soon as $p \leq q$. Finally, we show that thethree conjectures are in fact true for this superset.

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