Show that no integer u=4n+3 can be written as u = a^2 + b^2 where a,b are integers.
Hello,
Fact : For any integer m, $\displaystyle m^2 \equiv 0 \text{ or } 1 (\bmod 4)$
Proof :
A number m can only be even or odd.
If it's even, it can be written m=2k, in which case, $\displaystyle m^2=4k^2 \equiv 0 (\bmod 4)$
If it's odd, it can be written $\displaystyle m=2k+1$, in which case, $\displaystyle m^2=4k^2+4k+1 \equiv 1 (\bmod 4)$
Now, is it possible that $\displaystyle u=a^2+b^2 \equiv 3 (\bmod 4)$ when $\displaystyle a^2$ and $\displaystyle b^2$ can only $\displaystyle \equiv 0 \text{ or } 1 (\bmod 4)$ ?