NUMBERS REPRESENTED BY RESTRICTED SUMS OF FOUR SQUARES

Abstract

We prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. In particular, 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 $x+3y+3z$ is a power of 2.