Question

Find the number of non-zero integral solutions of the equation $|1-i{|}^x=2^x$.

Answer 1

Here $|1-i{|}^x=2^x$

$\Rightarrow [\sqrt{(1{)}^2+(-1{)}^2}{]}^x=2^x\Rightarrow (\sqrt{2}{)}^x=2^x$

$\Rightarrow 2^{\frac{x}{2}}=2^x\Rightarrow \frac{x}{2}=x$

$\Rightarrow \frac{x}{2}-x=0\Rightarrow \frac{-x}{2}=0$

$\Rightarrow x=0$

Thus the given equation has no non-zero integral solution.