This can be written (x-y)(x+y)=a^3. This implies that x-y=a and x+y=a^2 or x-y=a^2 and x+y=a. So 2x=a^2+a, making x=a(a+1)/2 and 2y=a^2-a, making y=a(a-1)/2; or x=a(a+1)/2 and y=a(1-a)/2. If a is odd a-1 and a+1 are both even, so divisible by 2. So x and y are integers because a(a-1) or a(a+1) are both divisible by 2.