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On the equivalence of cuboid equations and their factor equations

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.