Problem 3. a) Write the number 2016 as a sum of three perfect squares.

b) Show that the number $2016^{2015}$ can be written as the sum of three non-zero perfect squares.
Solution:
a) $2016=1600+400+16=40^{2}+20^{2}+4^{2} \ldots \ldots 2+4^{2}-36^{2}+12^{2}+b^{2}$

![](https://cdn.mathpix.com/cropped/2024_06_07_46238a177f184c489daag-1.jpg?height=71&width=1675&top_left_y=2187&top_left_x=183)

$N=\left(40^{2}+20^{2}+4^{2}\right) \cdot\left(2016^{1007}\right)^{2}=\left(40 \cdot 2016^{1007}\right)^{2}+\left(20 \cdot 2016^{1007}\right)^{2}+\left(4 \cdot 2016^{1007}\right)^{2} \ldots \ldots \ldots \ldots .3 p$

Note: Any other correct solution will be scored accordingly.