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.

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