Sums of two squares modulo 4

Let $m$ be an integer. If $m=2j$, then $m^2=4j^2\equiv0\pmod4$. If $m=2j+1$, then $m^2=4j(j+1)+1\equiv1\pmod4$. So a square is $\equiv0$ or $1\pmod4$.

A sum of two squares is therefore congruent to $0+0=0$, $0+1=1$ or $1+1=2$ modulo $4$, never to $3$. Hence $4k+3$ is never a sum of two squares.