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If $\displaystyle L = \frac{3+\sqrt{-3}}{2}$, and if $x\equiv 1\pmod{L}$,
show that $x^3\equiv 1\pmod{L^4}$.

I have already shown that if $x\equiv 1\pmod{L}$,
then $x^3\equiv 1\pmod{L^3}$.

Thanks.

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