Numbers represented by restricted sums of four squares

Abstract: In this paper, we prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. For example, we show that every nonnegative integer $n$ can be written as $x^{2}+y^{2}+z^{2}+t^{2}$ where $x,y,z,t$ are integers and $x+y+2z+2t$ is a square or a cube.