论文标题
班级号和佩尔方程$ x^2 + 105y^2 = z^2 $
Class Numbers and Pell's Equation $x^2 + 105y^2 = z^2$
论文作者
论文摘要
毕达哥拉斯三元曲线和椭圆曲线的两个良好研究的双方方程是第一个方程，第一个方程是我们通过单位圆上的有理点进行参数化的，第二个我们有一个理性溶液组的结构定理。最近，Yekutieli讨论了这两个问题之间的联系，并描述了毕达哥拉斯三元组的群体结构和给定斜边的三元组数量。在Arxiv：2112.03663中，我们将这些方法和结果概括为Pell的方程式。 We find a similar group structure and count on the number of solutions for a given $z$ to $x^2 + Dy^2 = z^2$ when $D$ is 1 or 2 modulo 4 and the class group of $\mathbb{Q}[\sqrt{-D}]$ is a free $\mathbb{Z}_2$ module, which always happens if the class number is at most 2. In this paper, we discuss the main ARXIV的结果：2112.03663在$ d = 105 $的情况下，使用一些具体示例。
Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently, Yekutieli discussed a connection between these two problems, and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. In arXiv:2112.03663 we generalized these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a given $z$ to $x^2 + Dy^2 = z^2$ when $D$ is 1 or 2 modulo 4 and the class group of $\mathbb{Q}[\sqrt{-D}]$ is a free $\mathbb{Z}_2$ module, which always happens if the class number is at most 2. In this paper, we discuss the main results of arXiv:2112.03663 using some concrete examples in the case of $D=105$.
