<s>[INST] Prove $6 \nmid [\left( \sqrt[3]{28} - 3 \right)^{-n}]$ Prove that: $$6 \not\left|\ \left\lfloor\frac 1 {(\sqrt[3]{28} - 3)^{n}}\right\rfloor \ (n \in Z^+)\right.$$
($\lfloor x\rfloor$ = largest integer not exceeding $x$)
I am very bad as English and number theory, please help me
 [/INST] If we set $\eta=\sqrt[3]{28}$ and $\omega=\dfrac1{\eta-3}=\dfrac{\eta^3-27}{\eta-3}=\eta^2+3\eta+9$, then, working $\bmod\ \eta^3-28$:
$$
\begin{align}
\omega^0&=1\\
\omega^1&=9+3\eta+\eta^2\\
\omega^2&=249+82\eta+27\eta^2\\
\omega^3&=6805+2241\eta+738\eta^2
\end{align}\tag{1}
$$
Solving the linear equations involved yields
$$
\omega^3-27\omega^2-9\omega-1=0\tag{2}
$$
Looking at the critical points of $x^3-27x^2-9x-1$, we see that it has one real root and two complex conjugate roots. The real root is $\omega\stackrel.=27.3306395$, and since the product of all the roots is $1$, the absolute value of the two conjugate roots is less than $\frac15$.
Let $\omega_0=\omega$ and $\omega_1$ and $\omega_2=\overline{\omega}_1$ be the roots of $x^3-27x^2-9x-1=0$. Symmetric functions and the coefficients of $(2)$ yield
$$
\begin{align}
a_0=\omega_0^0+\omega_1^0+\omega_2^0&=3\\
a_1=\omega_0^1+\omega_1^1+\omega_2^1&=27\\
a_2=\omega_0^2+\omega_1^2+\omega_2^2&=747\quad=27^2-2(-9)
\end{align}\tag{3}
$$
and, because each $\omega_k$ satisfies $(2)$,
$$
a_n=27a_{n-1}+9a_{n-2}+a_{n-3}\tag{4}
$$
Because $|\omega_1|=|\omega_2|<\frac15$, $|\,a_n-\omega^n\,|\le\frac2{5^n}$. Also, $(3)$ and $(4)$ show that $a_n\equiv3\pmod{6}$.
Therefore, $\omega^0=1$ and for $n\ge1$,
$$
\lfloor\omega^n\rfloor\in\{2,3\}\pmod{6}\tag{5}
$$
</s>