勾股三角形的不存在性及其在三维空间中的应用
A Non-Existence Property of Pythagorean Triangles with a 3-D Application
摘要：
在介绍之后，在第2节中，我们陈述了描述整个毕达哥拉斯三元组家族的众所周知的参数公式。在第3节中，我们列出了数论中的四个著名结果，这些结果将在后面的文章中使用。在第二节中，我们证明了两个命题。命题1说，丢番图方程z^2=x^4+4y^4，在正整数x，y和z中没有解。然后，我们使用命题1，证明了一个特定的四变量丢番图系统的不可解性，这是命题2。在第五节中，我们给出了一些有共同边界的勾股三角形对的例子。在第6节中，我们用命题2证明定理1：不存在一对勾股三角形，使得第一个三角形的最大边长是第二个三角形的边长;并且，第一个三角形的最小边长与第二个三角形的边长相同。在第7节中，我们介绍了有关勾股盒信息，并描述了如何生成无穷多个具有整数长度的面对角线的勾股盒。最后，在第8节中，我们应用命题2证明定理2：不存在一对全等（或相对）面为正方形的勾股盒。而在另一对相对的面上的四条等长的对角线也有整数长。
英文摘要：
After the introduction, in section 2 we state the well known parametric formulas that describe the entire family of Pythagorean triples. In section 3, we list four well known results from number theory, used later in the paper. in section 2, we prove two propositions. Proposition 1 says that the diophantine equation z^2=x^4+4y^4, has no solutions in positive integers x,y, and z. We then use Proposition 1, to prove the insolvability of a certain four-variable diophantine system, which is Proposition 2. In Section 5, we present some examplesof pairs of Pythagorean triangles with a common hypotenuse. In Section 6, we use Proposition 2 to prove Theorem 1: There exists no pair of Pythagorean triangles such that the leg of largest length in the first triangle is the hypotenuse of the second triangle; and in addition, with the leg of smallest length in the first triangle; having the same length as a leg in the second triangle. in Section 7, we present information on Pythagorean boxes, and we describe how to generate infinitely many Pythagorean boxes with a face diagonal of integral length. Finally, in Section 8, we apply Proposition 2 to prove Theorem 2: There exists no Pythagorean box with a pair of congruent (or opposite) faces being squares. And with the four diagonals of equal length in another pair of opposite faces, also having integral length.
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