不定方程x~2十y~2=m整数解的通解

本文利用Gauss整数环的性质,求出不定方程 x~2+y~2=m, (1)这里m为任意给定的正整数,整数解的通解,并由此求出不定方程(1)的解数r(m)的计算公式。r(m)的计算公式,即本文定理2,在华罗庚[1]第六章§7中已有结果,但证法不一样。

The General Solutions of Integers of the Indefinite Equations x~2+y~2=m

In this paper we use the properties of Gaussian integral ring to find the general solutions of integers of the indefinite equationx2 + y2 =m ( 1 )where m is an arbitrarily given integer, and furthermore to find the computational formula of the number r(m) of the solutions of integers of the equation (1). The formula of r(m) has been given in "The Introduction of Number Theory" by Hua Luogeng, but the proofs arc different.