A structure theorem for the restricted sum of four squares

Abstract

Let p be an odd prime. We show that each solution of the system of congruence equations $x_1^2+x_2^2+x_3^2+x_4^2\equiv 0\left(\mathrm{mod}p\right)$ and $x_1+x_2+x_3+x_4\equiv 0\left(\mathrm{mod}p\right)$ corresponds to precisely four solutions of the system of Diophantine equations $x_1^2+x_2^2+x_3^2+x_4^2=p^2$ and $x_1+x_2+x_3+x_4=p$ that are pairwise orthogonal over $Z$, partially answering a conjecture proposed in Wang et al. [10]. The result was obtained by counting the number of solutions of both equations using Gaussian sum and modular forms, and the classical Cayley transformation.