Perfect CuboidA perfect cuboid is a cuboid having integer side lengths, integer face diagonals
(1)
(2)
(3)
and an integer space diagonal
(4)
The problem of finding such a cuboid is also called the brick problem, diagonals problem, perfect box problem, perfect cuboid problem, or rational cuboid problem.
No perfect cuboids are known despite an exhaustive search for all "odd sides" up to 100 billion (10^11) (Weisstein, Eric W., "Perfect Cuboid" from MathWorld).
Solving the perfect cuboid problem is equivalent to solving the
Diophantine equations(5)
(6)
(7)
(8)
A solution with integer
space diagonal and two out of three face diagonals is
,
and
giving
,
,
and
, which was known to Euler. A solution giving integer space and face diagonals with only a single nonintegral
polyhedron edge is
,
and
, giving
,
,
and
.
via
mathworld.wolfram.com