  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.
