## Possible norms of Gaussian integers mod 4

Show that the norm of a Gaussian integer cannot be congruent to 3 mod 4.

Let $a+bi$ be a Gaussian integer. Recall that $N(a+bi) = a^2 + b^2$. Note that the squares mod 4 are 0 and 1; the possible sums of two squares mod 4 are 0, 1, and 2.