丢番图方程：一个系统的方法
Diophantine equations: a systematic approach
摘要：
本文开创了丢番图方程理论的一个新的研究方向：定义一个合适的方程大小，从最小的方程开始对所有的多项式丢番图方程进行排序，然后按此顺序求解。通过将一种新的计算机辅助程序与人类推理相结合，我们解决了大小小于$31$的所有多项式丢番图方程的希尔伯特第十问题，其中大小在（齐达内，2018）中定义。此外，我们解决了这个问题的所有方程的大小等于$31$，只有一个例外。此外，我们解决了希尔伯特的第十个问题的所有二元丢番图方程的大小小于$32$，所有对称方程的大小小于$39$，所有三个单项方程的大小小于$45$，并在每一个类别，确定明确的最小方程的问题仍然开放。结果，我们得出了一系列方程，这些方程写起来很简单，但显然很难求解。正如我们从费马大定理的例子中所知道的，这样的方程有可能刺激数论中新方法的发展。
英文摘要：
This paper initiates a novel research direction in the theory of Diophantine equations: define an appropriate version of the equation's size, order all polynomial Diophantine equations starting from the smallest ones, and then solve the equations in that order. By combining a new computer-aided procedure with human reasoning, we solved the Hilbert's tenth problem for all polynomial Diophantine equations of size less than $31$, where the size is defined in (Zidane, 2018). In addition, we solved this problem for all equations of size equal to $31$, with a single exception. Further, we solved the Hilbert's tenth problem for all two-variable Diophantine equations of size less than $32$, all symmetric equations of size less than $39$, all three-monomial equations of size less than $45$, and, in each category, identified the explicit smallest equations for which the problem remains open. As a result, we derived a list of equations that are very simple to write down but which are apparently difficult to solve. As we know from the example of Fermat's Last Theorem, such equations have a potential to stimulate the development of new methods in number theory.
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