great one

1) it has to be of the form 4k-1

what else?

When it is of the form 8k - 1 , we are sure that the sum is zero.

now then it is of the form 8k-5, then the real issue is...

The sum to me "seems not zero.. for any of those numbers" but that is more instinct based.

so to me the answer should be

8k-1!

(1+ix)^m

and

(1-ix)^m

then you will get

mCr x^r. i^r

and

mCr x^r. i^r (-1)^r

We take their sum and difference! and then their difference!!

I hope tihs is something useful and not a stupid invention of my mind! cos right now i am half awake.