Question

Is i^2=-1 for how many integers N is (N+i)^4 an integer?


so far i have factored out (N+i)^4 to be ... n^4+2n^2+1 i would suggest checking that, but from there i don't know where to go...doesn't it work for every integer?

Answer 1

first of all
(N+1)^4 = N^4 + 4N^3(i) + 6N^2(i^2) + 4N(i^3) + i^4
= N^4 + 4N^3(i) - 6N^2 - 4N(i) + 1
= N^4 + 4i(N^3 -N) + 1

for this to be an integer, the i-term must drop out, that is, N^3 - N = 0
N(N^2 - 1) = 0
N(N+1)(N-1) = 0
N = 0, -1, or +1

check my arithmetic