AI 中文总结
本文利用初等方法及相关数论结果，推广了T. H. Chan关于三个连续幂数的结论，讨论了若干丢番图方程的解。
AI 中文摘要
在T. H. Chan的近期工作中，已证明不存在三个连续幂数，其中间项为完全立方数，且另外两项各自仅含一个素因子的奇次幂。他们通过佩尔方程、椭圆曲线和二阶递推关系得到该结果。本文利用Nagell-Ljunggren、Lebesgue、Ko的结果及一些初等方法，将该结论推广至中间项可为任意次幂的无穷多情况。基于此结论，我们讨论了若干丢番图方程的解。
英文摘要
In the recent work by T. H. Chan, it was shown that there are no three consecutive powerful numbers with the middle term a perfect cube and each of the other two having only a single prime factor raised to an odd power. They get it with Pell equations, elliptic curves, and second-order recurrences. In this paper, with results given by Nagell-Ljunggren, Lebesgue, Ko and some elementary methods, we extend the result to similar cases, whose middle term can be perfect powers for infinitely many powers. With this conclusion, we discuss the solutions to some Diophantine equations.
