On Mordell's Equation
On Mordell's Equation
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关于莫德尔方程
DOI：
10.1023/a:1000281602647
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发表时间：
1998
影响因子：
1.8
通讯作者：
H. Zimmer
中科院分区：
文献类型：
--
作者：
J. Gebel;A. Pethö;H. Zimmer
在以前的论文中，我们提出了一个计算有理数Q上椭圆曲线上所有整点的算法。在这里，我们说明了我们的方法，将其应用到Mordell方程yk 0 k ∈ Z，并得出一些结论，从我们的数值结果。事实上，我们在Z中求解Mordell方程，对于范围0 <|K| [les ] 10 000，并将计算部分扩展到0 <|K|十万对于这些k值，霍尔猜想中的常数是C=5。其他一些有趣的意见是关于大整数点，大发电机的Mordell-Weil组和大泰特-Shafareviti组。三个图说明了整数点的分布与参数k的关系。一个有趣的特征是图中出现了线。
In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordell‘s Equation yk for 0 ≠ k ∈ Z and draw some conclusions from our numerical findings. In fact we solve Mordell‘s Equation in Z for all integers k within the range 0 < | k | [les ] 10 000 and partially extend the computations to 0 < | k | [les ] 100 000. For these values of k, the constant in Hall‘s conjecture turns out to be C=5. Some other interesting observations are made concerning large integer points, large generators of the Mordell–Weil group and large Tate–Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.