Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers $x$ and $y$ have to be determined. One non-elementary approach for this problem is the resolution via descent and class groups. Along these lines we formalized in Lean 3 the resolution of Mordell equations for several instances of $d<0$. In order to achieve this, we needed to formalize several other theories from number theory that are interesting on their own as well, such as ideal norms, quadratic fields and rings, and explicit computations of the class number. Moreover we introduced new computational tactics in order to carry out efficiently computations in quadratic rings and beyond.
翻译:分裂式方程式是数字理论中最受欢迎和活跃的研究领域。 在本文中,我们考虑的是莫德尔方程式,其形式为$y_2=x_3+d$,其中美元是一个(准)非零整数,所有解决方案均以美元和美元确定;这个问题的一个非基本办法是通过血统和阶级群体解决问题;按照这些方针,我们在利昂3中正式确定了Mordell方程式的解决方式,以支付几笔$d <0。为了实现这一目标,我们需要正式确定数字理论中其他一些理论,这些理论本身也很有趣,例如理想规范、四极田和环,以及明确计算等级数字。此外,我们引入了新的计算方法,以便在四极圈内外高效地进行计算。