Let $p$ be prime of the form $p ≡ 3 \pmod 4$. Show that $p$ cannot be written as the sum of two squares.
[Hint: Use the result of question 12 (b).]
We have shown that a square can only be 0 or 1 modulo 4.
So the sum of two squares can only be 0, 1 or 2 modulo 4, by Proposition (3.6).
And so any number congruent to 3 modulo 4 cannot be the sum of two squares.
Note $p$ doesn't need to be a prime, the result holds for any number congruent to 3 modulo 4.