具有原始面的毕达哥拉斯盒
Pythagorean Boxes with Primitive Faces
摘要：
Raymond A.Beauregard和E.R.Suryanarayan在他们的论文“Pythagorean Boxes”中定义了Pythagorean Rectangle的概念或概念，即边长和对角线长度都是整数的Pythagorean Rectangle（见[1]）;他们还引入了Pythagorean Box的概念，即边长和对角线长度都是整数的矩形三维平行六面体。同样在本文中，缩写概念PR将代表“毕达哥拉斯矩形”。在Beauregard和Suryanarayan的文章中，证明了存在无穷多个底为正方形、高为1的PB。本文给出了一个产生无穷多个PB的方法和公式，这些PB包含一对相反的（因此是全等的）PR是原始的;如果PR中包含的四个全等毕达哥拉斯三角形是本原的，则PR是本原的。本文有三个结果。在Result 1中，在结果2中，我们证明了如果满足类似的条件，那么可以生成无限多个包含一对（相反）PR的PB。在结果3中，我们证明了不存在具有正方形基底和一个面（因此是两个面）的PB。
英文摘要：
In their paper "Pythagorean Boxes", Raymond A.Beauregard and E.R.Suryanarayan define the concept or notion of Pythagorean Rectangle as one with sidelengths and integer diagonal lengths(see [1]);they also introduce the concept of a Pythagorean Box as a rectangular three-dimensional parallelepiped whose edges and diagonals have integer lengths.As in that paper, the abbreviation PB will simply stand for "Pythagorean Box";also in this article,the abbreviated notion PR will stand for "Pythagorean Rectangle". In the Beauregard and Suryanarayan paper,it was shown that there exist infinitely many PB's with a square base and height equal to 1.In this paper,we present a method and formulas that generate infinitely many PB's that contain a pair of opposite(and hence congruent)PR's which are primitive;a PR is primitive if the four congruent Pythagorean triangles contained there in are primitive.There are three results in this paper.In Result1,we derive certain explicit conditions that a PB must satisfy, if it possesses two pairs of(opposite)primitive PR's.In Result2,we show that if similar conditions are satisfied then infinitely many PB's can be generated containing a pair of(opposite)PR's.In Result3, we prove that there exist no PB's with a square base and a face(and hence two faces)which is a primitive PR.
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