# nt.number theory – Integer positive definite quadratic form in the form of sum of squares

Let $$A$$ to be symmetrical $$d times d$$ matrix with whole entries such as the quadratic form $$Q (x) = langle Axis, x rangle, x in mathbb {R} ^ d$$, is non-negative defined. For who $$d$$ does this imply that $$Q$$ is the sum of many squares of linear shapes with integer coefficients
$$Q (x) = sum_ {i = 1} ^ N ( ell_i (x)) ^ 2 quadric text {for some} , N?$$
For $$d = 2$$ That's right, I know about Sweden's 1995 IMO problem, but it's probably been known for a longer time.

I think I can prove it for other smaller, though less elemental dimensions (using Minkowski's theorem on lattice points in convex bodies: if $$Q$$ is positive definite, one can find a linear form $$ell (x)$$ such as $$Q- ell ^ 2$$ is always non-negative defined, it amounts to finding a whole point in an ellipsoid), but for the big ones $$d$$ this argument fails.