In [1] the authors constructed integral solutions of the Pythagorean equation x_1~2+...x_n~2+x__(n+1)~2=y~2, n≥1,(1) by using the identity (sum from i=1 to n of α_i~2)(sum from j=1 to n of α_j)~2+(sum from 1≤i≤j≤n α_i α_j)~2 (2) The point of this note is to observe that (2) can be rewritten as (2x_1)~2+...+(2x_n)~2+(||X||~-1)~=(||X||~2+1)~2. (3) where ||x||~2=x_1~2+...+x_n~2, and to show that (3) has an interesting geometric interpretation which is not apparent from (2).
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机译：在[1]中，作者使用等式（和）构造了勾股定方程x_1〜2 + ... x_n〜2 + x __（n + 1）〜2 = y〜2，n≥1，（1）的积分解。从i = 1到α_i〜2的n）（从j = 1到α_j的n）〜2 +（从1≤i≤j≤nα_iα_j的和）〜2（2）观察到（2）可以重写为（2x_1）〜2 + ... +（2x_n）〜2 +（|| X ||〜-1）〜=（|| X ||〜2 + 1）〜2 。 （3）其中|| x ||〜2 = x_1〜2 + ... + x_n〜2，并证明（3）具有有趣的几何解释，从（2）中看不到。
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