2020.11 – Expand That Binomial!

Given that i2=−1i^2 = -1, for how many integers is (n+i)4(n + i)^4 an integer?

Solution

Expanding gives

(n+i)4=n4+4n3i+6n2i2+4ni3+i4=n4−6n2+1+4n(n2−1)i. \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=4n(n+1)(n−1).0 = 4 n (n + 1)(n - 1).

So the answer is 3 (and the values are 0 and ± 1).