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? 

No, integers of the form $4n+3$ cannot be written as a sum of two squares.
To prove this, consider $z=x^2+y^2$ modulo $4$ and you'll see that you cannot get $3$.
