There is no general solution for `ninNN.`

For `n=0:`  

`(x+i)^0-(x-i)^0=0`

`1-1=0`

Which is valid for all real numbers hence `x inRR.`

For `n=1:`

`(x+i)^1-(x-i)^1=0`

`x+i-x+i=0`

`2i=0`

Which is never true hence there is no solution.

For `n=2:`

`(x+i)^2-(x-i)^2=0`

`x^2+2x i-1-(x^2-2x i-1)=0`

`4x i=0`

`x=0`

Obviously I could go on like this forever. E.g. for `n=3` solutions are `x_(1,2)=pm1/sqrt3.` But the point is that there are no general solution when `n` is an integer (or even worse a real number). Another way to prove this is to use de Moivre's formula.