Divisibility Biconditional Proof

Question

n $\in$ $\Bbb Z$. Prove that 2 $\vert$ (n$^4$ - 3) if and only if 4 $\vert$ (n$^2$ + 3).

Since this is an 'if and only if' proof, I know I have to prove both ways. What would the ways be? Also would I need a lemma?

Answer 1

$2 \mid (n^4-3)$ and $4 \mid (n^2 + 3)$ are true if and only if $n$ is odd. This can be shown in many ways, like contradiction to show that $n$ cannot be even.

We know that $4 \mid (n^2 + 3)$ if $n^2 \equiv 1 \pmod 4$, i.e. $n$ is odd. Since $n$ is odd, $2 \mid (n^4-3)$ is true.

For the converse, $2 \mid (n^4-3)$ is true if and only if $n^4 \equiv 1 \pmod 2$, implying $n$ is odd. This will imply that $4 \mid (n^2 + 3)$ is true.