Posted by car on Tuesday, January 18, 2011 at 9:08am.
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?

Math  Reiny, Tuesday, January 18, 2011 at 9:37am
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 iterm must drop out, that is, N^3  N = 0
N(N^2  1) = 0
N(N+1)(N1) = 0
N = 0, 1, or +1
check my arithmetic
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