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? 

For any integer n, n = 
0, 1, 2 or 3 (mod 4). So $n^{2}$ = 0 or 1 (mod 4). Then for
any integers a and b, $a^{2} + b^{2}$ = 0, 1 or 2 (mod 4). This means the sum of two squares can
only be in the form 4k, 4k+1 or 4k+2, but never 4k+3. Thus no integer of the form 4k+3
is the sum of two squares.
