On the existence of magic squares of powers

For any d 2, we prove that there exists an integer n₀ (d) such that there exists an n n magic square of dᵗh powers for all n n₀ (d). In particular, we establish the existence of an n n magic square of squares for all n 4, which settles a conjecture of V\'arilly-Alvarado. All previous approaches had been based on constructive methods and the existence of n n magic squares of dᵗh powers had only been known for sparse values of n. We prove our result by the Hardy-Littlewood circle method, which in this setting essentially reduces the problem to finding a sufficient number of disjoint linearly independent subsets of the columns of the coefficient matrix of the equations defining magic squares. We prove an optimal (up to a constant) lower bound for this quantity.