Can an integer of the form $4n+3$ written as a sum of two squares? Let $u$ be an integer of the form $4n+3$, where $n$ is a positive integer. Can we find integers $a$ and $b$ such that $u = a^2 + b^2$? If not, how to establish this for a fact? 

Let's assume x^2+y^2 = 4n+3, then either x or y has to be even. Let's assume x = 2z and write
(2z)^2+y^2 = 4n+3. This can also be written as follows: 
(2z+y)^2-4zy = 4n+3 by rearrangement we can write 
(2z+y)^2-1^2 =4n+2+4zy
(2z+y)^2-1^2=2(2n+2zy+1) and further 
(2z+y-1)(2z+y+1)=2(2n+2zy+1)
The left side is product of two even numbers the right side is the product of even and odd number.  So the assumption is wrong, and no number in the form of 4n+3 can be a sum of two squares.
