A structure theorem for the restricted sum of four squares.

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 ≡ 0 (mod p) and x 1 + x 2 + x 3 + x 4 ≡ 0 (mod p) 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. [ABSTRACT FROM AUTHOR]