Benjamin D. Sokolowsky, Amy G. Vanhooft, Rachael M. Volkert, Aand Clifford A. Reiter, An Infinite Family Of Perfect Parallelepipeds, Mathematics of Computation, 83 (2014) 2441-2454.

A perfect parallelepiped has edges, face diagonals, and body diagonals all of integer length. We prove the existence of an infinite family of dissimilar perfect parallelepipeds with two nonparallel rectangular faces. We also show that we can obtain perfect parallelepipeds of this form with the angle of the nonrectangular face arbitrarily close to 90 degrees. Finally, we discuss the implications that this family has on the famous open problem concerning the existence of a perfect cuboid. This leads to two conjectures that would imply no perfect cuboid exists.

See Also: · [Abstract] for "Perfect Parallelepipeds Exist". · [Materials] for "Perfect Parallelepipeds Exist". · [Abstract][Preprint] for related manuscript "Pursuing the Perfect Parallelepiped". · [Abstract][Preprint] for related manuscript "Matrix generation of the Diophantine Solution of 3 ≤ n ≤ 9 Squares that are Square". · [Abstract][Preprint] for related manuscript "Families of Nearly Perfect Parallelepipeds".