"Exponential Madness" (Gauss's challenge) From Euler's identity, we see that 

$e^{i\pi}=-1$

$\Rightarrow e^{2ik\pi}=1$ [squaring both sides]. 

This equation surely holds for all integers $k$.

EDIT: From the second equation we get $e^{1+i2k\pi}=e$

So, $e^{1+i2k\pi}=\left (e^{1+i2k\pi}\right )^{(1+i2k\pi)}$

and $e=e^{(1+i2k\pi)^2}=e^{1+i4k\pi}e^{-4k^2\pi^2}$  $ ...(*)$

But $e^{i4k\pi}=1$ from Euler's identity and $e^{1+i4k\pi}=e$

So $e=ee^{-4k^2\pi^2}$ and hence $e^{-4k^2\pi^2}=1$   $[$from $(*)$$]$

But this is true only for $k=0$ whereas we started out with a statement true for all $k\in \mathbb{Z}$. What is the catch here?
NOTE: 

$$(x^k)^2 \not = x^{k^2}, \text{ instead} \quad (x^k)^2 = x^{2k}$$



For the second update note that the summation/ product of  exponents is valid only for real exponent. In other words $(e^{1+2ki\pi})^{1+2ki\pi} \not = e^{(1+2ki\pi)^2}$
Thank you for an entertaining and instructive paradox. Most of your humourless commenters have missed the point, because the point involves the use of reason rather than faith in scriptural authority. 

The catch is that the only values of $z$ for which $e^z$ has one value are those for which the real part of $z$ is an integer. Thus $$e^{(1-4k^2\pi^2)+4k\pi{i}}$$ is single-valued only when $1-4k^2\pi^2$ is an integer: that is, when $k=0$.

I am not going to quote formulae to prove this, because formula don't provide truth, only reflect it. But here is some motivation. 


$1^{1/1}$ has just one value: $1$.
$1^{1/2}$ has two values: $1, -1$.
$1^{1/4}$ has four values: $1, -1, i, i$.
$1^{1/n}$ has $n$ values. 


For irrational exponents $x$, $1^x$ has infinitely many values, one of which is $1$.

As long as a family of values all behave in the same way, this multi-valuedness doesn't much matter. What your paradox has done is point out a sequence of operations where the multiplicity of the values causes trouble if you ignore it. Well done for pointing it out! To be brutal about it, not many people would think that (for example) $1^\pi$ has infinitely many values.  (In general they all have the form $e^{2k\pi^2{i}}$ for some integer $k$.