关于长方体方程及其因子方程的等价性
On the equivalence of cuboid equations and their factor equations
摘要：
欧拉长方体是具有整数边和整数面对角线的矩形平行六面体。如果欧拉长方体的空间对角线也是整数，则称其为完美长方体。一些欧拉长方体已经被发现。至于完美长方体，目前还没有人知道，也没有证明它们的不存在。欧拉长方体和完美长方体由某些丢番图方程系统描述。这些方程具有固有的$S_3$对称性。最近，他们因式分解关于这个$S_3$对称性和因子方程推导。在本文中的因子方程是等价的，以原长方体方程的搜索完美的长方体和选择欧拉长方体。
英文摘要：
An Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals. An Euler cuboid is called perfect if its space diagonal is also integer. Some Euler cuboids are already discovered. As for perfect cuboids, none of them is currently known and their non-existence is not yet proved. Euler cuboids and perfect cuboids are described by certain systems of Diophantine equations. These equations possess an intrinsic $S_3$ symmetry. Recently they were factorized with respect to this $S_3$ symmetry and the factor equations were derived. In the present paper the factor equations are shown to be equivalent to the original cuboid equations regarding the search for perfect cuboids and in selecting Euler cuboids.
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