论文标题
在马尔可夫号码上：固定分子，分母和总和
On the Markov numbers: fixed numerator, denominator, and sum conjectures
论文作者
论文摘要
Markov数字是Diophantine方程的正整数解决方案$ x^2 + y^2 + z^2 = 3xyz $。马尔可夫已经在1880年表明所有这些解决方案都可以沿二进制树生成。因此，通过0到1之间的理由索引马尔可夫数字，这是相当平常的（而且很有用），该理性位于船尾式二进制树中的同一位置。 Frobenius的猜想声称每个马尔可夫号码最多出现在树上。 特别是，如果猜想是正确的，那么马尔可夫号码的顺序将在理由上建立新的严格顺序。 Aigner提出了三个猜想，以更好地理解这一顺序。第一个已经解决了几个月。我们证明了其他两个猜想也是正确的。 一路上，我们将马尔可夫号码概括为非负整数的任何夫妇（P，Q）（不仅是在相对较高的素数时），并猜测，只要$ p \ p \ leq q $，统一性仍然是正确的。最后，我们证明了这三个猜想实际上是这个超集的。
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.
