I just don't understand where the confusion comes from. If you find x_i, y_i. s.t. x_i^2+y_i^2+1 = 0 (mod p_i), then use the Chinese Remainder Theorem to find x and y, then x=x_i (mod p_i), y=y_i (mod p_i). So x^2+y^2+1 = x_i^2+y_i^2+1 (mod p_i)
I just don't understand where the confusion comes from. If you find x_i, y_i. s.t. x_i^2+y_i^2+1 = 0 (mod p_i), then use the Chinese Remainder Theorem to find x and y, then x=x_i (mod p_i), y=y_i (mod p_i). So x^2+y^2+1 = x_i^2+y_i^2+1 (mod p_i)