2020.11 – Expand That Binomial!

Expanding gives

$$\begin{aligned}(n+i)^4 &= n^4 + 4 n^3 i + 6 n^2 i^2 + 4 n i^3 + i^4 \\&= n^4 - 6 n^2 + 1 + 4 n (n^2 - 1) i\end{aligned}$$

This is an integer if and only if the imaginary part is zero, i.e., when $0 = 4 n (n + 1)(n - 1)$.

So the answer is $3$ (and the values are $0$ and $\pm 1$).