$\textit{Why}$ the cases in 4 $\nmid n^2 - 3?$ (bases cases in modular induction) This question has already been answered here. My question is, when we suppose the hypothesis is not true (proof by contradiction), namely that $4 \mid (n^2 - 3)$, we get $n^2 - 3 = 4k$ for some integer $k$ and then we have two cases. But where does the two cases come from? In addition, why does putting them into two cases work to prove the statment? My thought process would be to try to prove this statement by contradiction and somehow figure out a way to produce a contradiction similar to how $\sqrt{2}$ is proved by contradiction. Any help is great!
Let me know if I need to clarify more!

So $n^2\equiv 0,1\pmod 4$. Hence $n^2-3\equiv 1,2\pmod 4$ which is not a multiple of $4$.
