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

1980 AMC 12 Problem 17

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17.

Given that

i^2=-1,

for how many integers

n

(n+i)^4

an integer?

none

1

2

3

4

Expand

(n+i)^4

and isolate its imaginary part

An integer has imaginary part zero, so solve the resulting cubic factorization

Expansion gives

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

The imaginary part vanishes exactly when

n=0,

n=1,

n=-1.

For each of these three integers the real part is an integer, so there are

3

values.

Therefore, the correct answer is D.