Show that no integer u=4n+3 can be written as u = a^2 + b^2 where a,b are integers.

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- May 1st 2009, 05:48 AMmpryalintegers
Show that no integer u=4n+3 can be written as u = a^2 + b^2 where a,b are integers.

- May 1st 2009, 05:59 AMMoo
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 $\displaystyle \equiv 0 \text{ or } 1 (\bmod 4)$ ?