Can $x^2+y^2,y^2+z^2,z^2+x^2$ and $x^2+y^2+z^2$ all be square numbers? I know that if we want $x^2+y^2$ to be square number, we are looking for pythagorean triple; if we want $x^2+y^2+z^2$ to be a square number, we are looking for pythagorean quadruple. But have we ever found any positive integers $x,y,z$ such that $x^2+y^2,y^2+z^2,z^2+x^2,x^2+y^2+z^2$ are all square numbers?

It's a famous open problem : the perfect cuboid problem. 
