Find the number of non-zero integral solutions of the equation $|1-i|^{x}=2^{x}$
Find the number of non-zero integral solutions of the equation $|1-i|^{x}=2^{x}$
$|1-i|^{x}=2^{x}$
$\Rightarrow(\sqrt{1^{2}+(-1)^{2}})^{x}=2^{x}$
$\Rightarrow(\sqrt{2})^{x}=2^{x}$
$\Rightarrow 2^{x / 2}=2^{x}$
$\Rightarrow \frac{x}{2}=x$
$\Rightarrow x=2 x$
$\Rightarrow 2 x-x=0$
$\Rightarrow x=0$
Thus, $0$ is the only integral solution of the given equation. Therefore, the number of nonzero integral solutions of the given equation is $0 .$
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