Proof that b is not divisible by 6

$$b=\left \lfloor (\sqrt{28}-3)^{-n} \right \rfloor$$

The brackets mean that the number is the largest integer smaller than $(\sqrt{28}-3)^{-n}$

Proof that b is never divisible by 6.

I have no idea how to even start, I thought to do it by induction, but that didn't even remotely work :(

Anybody keen to help out

• You claim this is contest-math. Source? Dec 2 '13 at 22:23

$$(\sqrt{28} - 3)^{-n} + (\omega \sqrt{28} - 3)^{-n} + (\omega^2 \sqrt{28} - 3)^{-n}$$
is an integer, where $\omega$ is a primitive cube root of unity.
Note that $3=\sqrt{27}$, so $\sqrt{28}-3=\frac{28-27}{\sqrt{28^2}+\sqrt{28\cdot 27}+\sqrt{27^2}}$, so we are looking at $\left\lfloor \left( \sqrt{28^2}+\sqrt{28\cdot 27}+\sqrt{27^2}\right)^n\right \rfloor$ Now experimentation says the number inside the floor is just a tiny bit less than a multiple of $3$ for $n \gt 1$. Usually this would be proved by finding a $c$ such that $\left( \sqrt{28^2}+\sqrt{28\cdot 27}+\sqrt{27^2}\right)^n + c^n$ is an integer and a multiple of $3$ and $|c| \lt 1$
• Shouldn't it be ${\color{red}+}\sqrt{28\cdot 27}$? Dec 2 '13 at 21:58