关于Hajdu和Tengely的一个问题
On a problem of Hajdu and Tengely
摘要：
我们证明了Hajdu和Tengely在最近的一篇论文中完成的关于由平方和五次幂组成的本原等差数列的研究的一个结果：唯一的互质整数形式为（a^2，b^2，c^2，d^5）的等差数列是（1，1，1，1）.为了证明，我们首先将问题归结为确定亏格为4的三条特定超椭圆曲线上的有理点集的问题。一个2-覆盖下降计算表明，有两个这些曲线上没有有理点。我们找到了最后一条曲线的Mordell-Weil群的有限指数子群的生成元。应用Chabauty方法，我们证明了这条曲线上仅有的有理点是明显点.
英文摘要：
We prove a result that finishes the study of primitive arithmetic progressions consisting of squares and fifth powers that was carried out by Hajdu and Tengely in a recent paper: The only arithmetic progression in coprime integers of the form (a^2, b^2, c^2, d^5) is (1, 1, 1, 1). For the proof, we first reduce the problem to that of determining the sets of rational points on three specific hyperelliptic curves of genus 4. A 2-cover descent computation shows that there are no rational points on two of these curves. We find generators for a subgroup of finite index of the Mordell-Weil group of the last curve. Applying Chabauty's method, we prove that the only rational points on this curve are the obvious ones.
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