Arithmetic behaviour of the sums of three squares

Let n ≠ 4^a(8b + 7) be an integer. We deal with the problem of the solvability of the equation n = x₁² + x₂² + x₃² in integers x₁, x₂, x₃ prime to n. By a theorem of Vila (Arch. Math. 44 (1985), 424–437), the existence of such a solution implies that every central extension of the alternating group A_n, for n ≡ 3 (mod 8), can be realized as a Galois group over Q.