A 'Perfect Cuboid' (also called a Perfect Euler Brick) is a (proposed) cuboid whose space diagonal also has integer length. That's to say a² + b² + c² = g² (where a, b, c are the sides and g is the diagonal).

The problem of their existence (or not) goes back at least as far as mathematiciian Paul Halcke, 300 years or so ago.

There is currently no generally accepted mathematical proof to determine whether they are possible physical structures or not. Computer modelling searches (running to trillions of examples) have so far failed to find one.

Further reading : Wikipedia