I'm working on the following two proofs: 1.) (x+y)2n+1 = x2n+1 + y2n+1 if and only if x=0, y=0 or y=-x and 2.) (x+y)2n = x2n + y2n if and only if x=0 or y=0 I've tried using induction and get stuck at a certain point. I've also tried playing around with summations since there is a binomial expansion in there. I haven't had any luck. Earlier today however I found that this equality is true: x2n+1 + y2n+1 = (x+y)(x2n - x2n-1y + x2n-2y2 - ... - xy2n-1 + y2n) I haven't played around with this yes, but I think setting this equal to (1) might help prove this. I'm just wondering if anyone has any ideas on this.