Treasure Hunting Perfect Euler bricks

An Euler brick is a cuboid with integer side dimensions such that the face diagonals are integers. Already in 1740, families of Euler bricks have been found. Euler himself constructed more families. If the space diagonal of an Euler brick is an integer too, an Euler brick is called a perfect Euler brick. Nobody has found one. There might be none. Nevertheless, it is an entertaining sport to go for this treasure hunt for rational cuboids and search of course with the help of computers. We especially look in this lecture at Soundersons parametrizations and give a short proof of a theorem of Spohn telling that the any of these Euler bricks is not perfect. But there are other parameterizations. Introduction: the map of John Flint An Euler brick is a cuboid of integer side dimensions a, b, c such that the face diagonals are integers. If u, v, w are integers satisfying u + v = w, then the Sounderson parametrization (a, b, c) = (|u(4v − w)|, |v(4u − w)|, |4uvw|) leads to an Euler brick.