We can factor the base of 2007 as follows:
2007 =3^2 x 223
Now we can raise the above 2 factors to the power of 2007:
(3^2)^2007 x 223^2007
3^4014 x 223^2007. So from the 4014 exponent, we have 4014/2 =2007 Perfect Squares. And from the 2007 exponent, we have: 2007 -1 / 2 =1,003 Perfect Squares. Then we multiply them together, we get: 2007 x 1003=2,013,021 Perfect Squares. To this will add the sum of the same number of perfect squares, i.e., 2007 + 1003 =3010 Perfect Squares.
So the final total =[1,003 x 2,007] + 3,010 =2,016,031 in total.
And that is my attempt !!!.
P.S. I neglected to take "1" as a perfect square !!. Therefore, if we add 1 to the above total, we have: 2,016,031 + 1 =2,016,032 !!.