Earlier this month (December 1), the 62nd William Lowell Putnam Mathematical Competition was held at universities throughout the United States and Canada. In commemoration, this month's problem is from a previous Putnam.

Show that the equation *x*^{2} - *y*^{2} =
*a*^{3} always has integer solutions for *x* and
*y* whenever *a* is a positive integer.

**Source: 1954 Putnam Competition**