On the Representation of the 1ntegees as a Difference of Squares

In recent times a number of authors (see [l]-[4]) have revisited the well-known results of Fermat and Jacobl In connection with the polygonal representation of the integers. In the papers cited an. alternate derivation for such formulas giving the total number of representations of an integer as the sum of either two triangular or square numbers was provided. These enumerations, which are given in terms of elementary divisor functions, were deduced as a consequence of the Gauss-Jacobi triple product identity. In contrast to sums of polygonal numbers, the author has investigated within [5] the representation of the integers as a difference of two triangular numbers. By use of a purely combinatorial argument, it was shown that the number of such representations of an integer n was exactly equal to the number of odd divisors of n. In this note we propose to extend the methods employed in [5] to the case of squares to prove the following result.