Diophantine Brick

Is there a three-dimensional rectangular parallelepiped, whose lengths of all edges and face diagonals of the space diagonal are integers? This task at least 300 years. Classical methods of the theory of Diophantine equations is proved that such a parallelepiped does not exist. Along the way, the hypothesis was confirmed by Euler that the minimum parallelepiped whose edges are intact, and face diagonally (but not spatial diagonal) is a parallelepiped with edges 44, 117, 240 and facial diagonals 125, 267, 244.