Jorge F. Sawyer and Clifford A. Reiter, Perfect Parallelepipeds Exist, Mathematics of Computation, 80 (2011) 1037-1040.

There are parallelepipeds with edge lengths, face diagonal lengths and body diagonal lengths all positive integers. In particular, there is a paral- lelepiped with edge lengths 271, 106, 103, minor face diagonal lengths 101, 266, 255, major face diagonal lengths 183, 312, 323, and body diagonal lengths 374, 300, 278, 272. Focused brute force searches give dozens of primitive perfect parallelepipeds. Examples include parallellepipeds with up to two rectangular faces.

See Also: · [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".