Sums of squares of integral linear forms

In this paper we prove that every positive definite n-ary integral quadrati c form with 12 less than or equal to n less than or equal to 13 (respective ly 14 less than or equal to n less than or equal to 20) that can be represe nted by a sum of squares of integral linear forms is represented by a sum o f 2 . 3(n) + n + 6 (respectively 3 . 4(n) + n + 3) squares. We also prove t hat every positive definite 7-ary integral quadratic form that can be repre sented by a sum of squares is represented by a sum of 25 squares.