Exercise 2.1.12 ($19$ is not a divisor of $4 n^2 + 4$ for any $n$)

Proof. Assume for contradiction that $19\mid 4n^2+4={(2n)}^2+2^2$.

Since $19\equiv 3(\mathit{mod}4)$, Lemma 2.14 implies that $19\mid 2$. This is false, so $19$ is not a divisor of $4n^2+4$ for any $n$. □