What's wrong with this?

$\displaystyle {\displaystyle e^{i2\pi n}=1 \text{ ; for } n=0,\pm 1, \pm 2, \hdots }$

$\displaystyle \displaystyle {ee^{i2\pi n}=e=e^{1+i2\pi n} }$

$\displaystyle e^{1+i2\pi n}=\left\{ e^{1+i2\pi n} \right\}^{(1+i2\pi n)}$

$\displaystyle e=e^{(1+i2\pi n)^2}=e^{1+i4\pi n-4\pi^2n^2}=e^{1+i4\pi n}e^{-4\pi^2n^2}$

$\displaystyle e^{1+i4\pi n}=e$

$\displaystyle e=ee^{-4\pi^2n^2}$

$\displaystyle 1=e^{-4\pi^2n^2}$, which is true only for $\displaystyle n=0$.