A Partial Residue Categorization of the Magic Square of Squares

It is unknown at present whether a magic square of squared integers exists. Such an object is defined to be a 3 by 3 grid of 9 distinct integer squares, such that the entries of each row, column, and two main diagonals sum to the same total. We give a categorization of the residue of the magic square of squares for primes dividing the central entry. The categorization leads to a magic square of square solution over a finite field and, lastly, some conjectures are made regarding the existence of certain residues.