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 from $[0,1]$ 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 $(p,q)\in {\mathbb{N}}^2$ (not only when they are relatively prime) and conjecture that the unicity is still true as soon as $p\le q$. Finally, we show that the three conjectures are in fact true for this superset.

马尔可夫数是Diophantine方程的正整数解 $X^{\mathrm{2个}}+{ÿ}^{\mathrm{2个}}+{ž}^{\mathrm{2个}}=3Xÿž$。早在1880年，马尔可夫就证明所有这些解决方案都可以沿着二叉树生成。因此，根据$[0，\mathrm{1个}]$在Stern-Brocot二叉树中的同一位置。弗罗贝尼乌斯（Frobenius）的猜想声称，每个马尔可夫数在树中最多出现一次。

特别是，如果猜想是正确的，则马尔可夫数的阶数将在有理数上建立新的严格阶数。艾格纳提出了三个猜想，以更好地理解这个顺序。第一个已经解决了几个月。我们证明其他两个猜想也是正确的。

一路上，我们将马尔可夫数推广到任何一对 $（p，q）\in {\mathbb{ñ}}^{\mathrm{2个}}$ （不仅当它们是相对质数的时候），而且推测唯一性一经成立就仍然是正确的 $p\le q$。最后，我们证明这三个猜想对这个超集实际上是正确的。