Research on pythagorean equation when given positive integer N, has the equation a solution, how many solutions are there, and how to solve it. Prove the existence one-to-one corresponding relationship between the solution of N, the basic solution and the primitive solution of the factor of N. Then using the primitive solution of prime number and pythagorean plot operation of two groups of the primitive solution, by means of successive recursion, derives the methods to calculate the primitive solution of different types factor of N, which obtained all solutions of arbitrary N, and obtained elementary formula to calculate of the number of all group solutions. The given algorithm also filled the blank of research of pythagorean number over the years.% 研究勾股方程给定正整数N 时，方程是否有解，有几组解，怎样求解。在证明N的解与N 的因数的基本解和本原解三者之间存在着一一对应关系的基础上，利用素数的本原解和两组本原解的勾股积运算，经逐次递推，导出了计算N 的各种不同类型因数的本原解的计算方法，得到了计算任意N 的所有解的简捷方法，并给出了计算全部解的组数的初等公式。填补了多年来研究勾股数的一个空白。
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