Positive integers $n$ with $(1+i)^n=(1-i)^n$

Characterize positive integers $n$ such that $(1+i)^{n}=(1-i)^{n}$

Solution

Since $1-i\ne0$, the desired equality is equivalent to \[ \left(\frac{1+i}{1-i}\right)^n=1. \] But \[ \frac{1+i}{1-i}=i, \] so the condition is $i^n=1$. The powers of $i$ have period $4$, hence \[ \boxed{4\mid n}. \] Thus the positive integers with the required property are exactly $n=4,8,12,\dots$.