关于不定方程x^2+py^2=z^2解的存在性

Existence of solutions for Diophantine equation x^2+py^2=z^2

利用初等的方法,证明不定方程x^2+py^2=z^2存在无数多组解,给出了解的结构,并给出了一些特殊解.这为求不定方程x^2+py^2=z^2的正整数解提供了一个切实有效的公式化方法.根据不定方程x^2+py^2=z^2解的结构,给出了对应的二次曲线x^2+py^2=1上有理点的个数及结构.

By using elementary method,it is proved that the Diophantine equation x^2+py^2=z^2 has innumerable groups of solutions.The structure of the solutions and some special solutions was given.This provides a practical and effective formulaic method for finding positive integer solutions of Diophantine equation x^2+py^2=z^2.According to the structure of the solution of Diophantine equation x^2+py^2=z^2,the number and structure of rational points on the corresponding conic x^2+py^2=1was given.